Chapter 6 Integers (Additional Questions)

An integer is a whole number (not a fractional number) that can be positive, negative, or zero. The set of integers is often denoted by $\mathbb{Z}$ and includes numbers like ..., -3, -2, -1, 0, 1, 2, 3, ...

Let's examine each option:

(A) $\frac{1}{2} = 0.5$

This is a fraction or a decimal with a fractional part. It is not a whole number.

(B) $-5$

This is a negative whole number. It fits the definition of an integer.

(C) $2.5$

This is a decimal with a fractional part. It is not a whole number.

(D) $\sqrt{3}$

This is an irrational number, approximately $1.732...$. It is not a whole number.

Based on the definition and analysis of each option, only $-5$ is an integer.

The correct option is (B).

In mathematics and everyday representation, temperatures above zero are represented by positive numbers, and temperatures below zero are represented by negative numbers.

The phrase "$10^\circ\text{C}$ below zero" means a temperature that is 10 units less than zero.

This can be expressed as $0 - 10$.

Therefore, the integer representing a temperature of $10^\circ\text{C}$ below zero is $-10$.

Comparing this with the given options:

(A) $10$ represents $10^\circ\text{C}$ above zero.

(B) $-10$ represents $10^\circ\text{C}$ below zero.

(C) $0$ represents $0^\circ\text{C}$, which is the zero point.

(D) $20$ represents $20^\circ\text{C}$ above zero.

The integer representing a temperature of $10^\circ\text{C}$ below zero is $-10$.

The correct option is (B).

A number line is a straight line with zero at some point, with positive numbers to one side and negative numbers to the other.

In a standard horizontal number line:

The point representing zero (0) is typically placed in the center.

Numbers greater than zero (positive numbers) are placed to the right of zero.

Numbers less than zero (negative numbers) are placed to the left of zero.

Positive integers are the integers $1, 2, 3, 4, ...$. Since these numbers are greater than zero, they are located to the right of 0 on a horizontal number line.

Therefore, to represent positive integers from 0 on a number line, you move to the Right.

Comparing this with the given options:

(A) Left: This is the direction for negative numbers from zero.

(B) Right: This is the direction for positive numbers from zero.

(C) Up: This direction is typically used on a vertical number line (like a thermometer scale) to represent increasing values (positive direction).

(D) Down: This direction is typically used on a vertical number line to represent decreasing values (negative direction).

For a standard number line (horizontal), the direction for positive integers from 0 is to the right.

The correct option is (B).

In financial contexts or when representing changes, we often use integers to denote gains and losses.

A profit signifies a gain or an increase in value. Gains are conventionally represented by positive integers.

A loss signifies a decrease in value. Losses are conventionally represented by negative integers.

Zero represents no change (neither profit nor loss).

The question asks for the integer representing a profit of $\textsf{₹}$ 500.

Since it is a profit, which is a gain, it should be represented by a positive integer.

The magnitude of the profit is 500.

Therefore, the integer representing a profit of $\textsf{₹}$ 500 is $+500$, which is usually written simply as $500$.

Comparing this with the given options:

(A) $-500$ represents a loss of $\textsf{₹}$ 500.

(B) $500$ represents a profit of $\textsf{₹}$ 500.

(C) $0$ represents neither profit nor loss.

(D) $1000$ represents a profit of $\textsf{₹}$ 1000.

The integer representing a profit of $\textsf{₹}$ 500 is $500$.

The correct option is (B).

Integers can be classified into three main categories:

• Positive integers: These are integers greater than zero ($>$ 0), such as $1, 2, 3, ...$
• Negative integers: These are integers less than zero ($<$ 0), such as $-1, -2, -3, ...$
• Zero: This is the integer $0$.

By definition, positive integers are strictly greater than $0$, and negative integers are strictly less than $0$.

The integer $0$ is neither strictly greater than $0$ nor strictly less than $0$.

Therefore, $0$ is the integer that is neither positive nor negative.

Comparing this with the given options:

(A) $1$ is a positive integer.

(B) $-1$ is a negative integer.

(C) $0$ is neither positive nor negative.

(D) Any positive integer is, by definition, positive.

The integer that is neither positive nor negative is $0$.

The correct option is (C).

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number.

The absolute value of a number $x$ is denoted by $|x|$.

The definition of absolute value is:

$|x| = x$, if $x \geq 0$

$|x| = -x$, if $x < 0$

We need to find the absolute value of $-15$.

Here, the number is $-15$. Since $-15 < 0$, we use the second part of the definition: $|x| = -x$ for $x < 0$.

So, for $x = -15$, $|-15| = -(-15)$.

The negative of a negative number is a positive number.

Therefore, the absolute value of $-15$ is $15$.

Comparing this with the given options:

(A) $-15$: This is the original number, which is negative.

(B) $15$: This is the positive value corresponding to the distance from zero.

(C) $0$: This is the absolute value of $0$, not $-15$.

(D) $-5$: This is not related to the absolute value of $-15$.

The absolute value of $-15$ is $15$.

The correct option is (B).

To compare two integers, we can think about their positions on a number line. On a horizontal number line, numbers increase as you move from left to right.

Consider the integers $-7$ and $-3$. Let's visualize their positions relative to $0$ on a number line:

... -7 \$ -6 \$ -5 \$ -4 \$ -3 \$ -2 \$ -1 \$ 0 \$ 1 \$ 2 ...

As you move from left to right, the numbers get larger.

Looking at the number line, $-7$ is located to the left of $-3$.

Since $-7$ is to the left of $-3$ on the number line, $-7$ is less than $-3$.

Alternatively, for negative numbers, the number with the smaller absolute value is the larger number.

Absolute value of $-7$ is $|-7| = 7$.

Absolute value of $-3$ is $|-3| = 3$.

Since $3 < 7$, the number with the absolute value $3$ (which is $-3$) is greater than the number with the absolute value $7$ (which is $-7$) when both are negative.

Comparing this with the given options:

(A) $-7 > -3$: This inequality states that $-7$ is greater than $-3$, which is false.

(B) $-7 < -3$: This inequality states that $-7$ is less than $-3$, which is true.

(C) $-7 = -3$: This states that $-7$ is equal to $-3$, which is false.

(D) Cannot compare: We can definitely compare integers.

The correct comparison is $-7 < -3$.

The correct option is (B).

The given set of integers is $\{-10, 0, 5, -2\}$.

To find the smallest integer in the set, we need to compare the values of the integers provided.

We have positive integers ($5$), zero ($0$), and negative integers ($-10, -2$).

Recall that on a horizontal number line, numbers increase in value as you move from left to right. Numbers to the left are smaller than numbers to the right.

The order of these numbers on a number line is:

... -10 \$ ... \$ -2 \$ ... \$ 0 \$ ... \$ 5 \$ ...

From this visualization, we can see the relative positions:

$-10$ is to the left of $-2$. So, $-10 < -2$.

$-2$ is to the left of $0$. So, $-2 < 0$.

$0$ is to the left of $5$. So, $0 < 5$.

Combining these comparisons, the order from smallest to largest is:

$-10 < -2 < 0 < 5$

Alternatively, we know that:

• Any positive integer is greater than $0$ and any negative integer.
• $0$ is greater than any negative integer.
• Among negative integers, the one with the larger absolute value is smaller.

In the set $\{-10, 0, 5, -2\}$:

$5$ is the only positive integer, so it is the largest.

$0$ is greater than the negative integers ($-10$ and $-2$).

We need to compare the negative integers $-10$ and $-2$.

The absolute value of $-10$ is $|-10| = 10$.

The absolute value of $-2$ is $|-2| = 2$.

Since $10 > 2$, and both are negative numbers, $-10$ is smaller than $-2$.

Thus, the smallest integer in the set $\{-10, 0, 5, -2\}$ is the smallest among the negative integers, which is $-10$.

Comparing our finding with the given options:

(A) $-10$

(B) $0$

(C) $5$

(D) $-2$

The smallest integer is $-10$.

The correct option is (A).

Descending order means arranging numbers from the largest value to the smallest value.

The given integers are: $6, -4, 0, -8, 2$.

To arrange them in descending order, we need to compare their values.

We can classify the integers:

• Positive integers: $6, 2$
• Zero: $0$
• Negative integers: $-4, -8$

Recall that:

• Positive integers are always greater than zero and negative integers.
• Zero is greater than any negative integer.
• Among negative integers, the integer closer to zero (i.e., with a smaller absolute value) is the larger integer.

Let's arrange the given integers from largest to smallest:

1. The largest are the positive integers: $6$ and $2$. Comparing them, $6 > 2$. So, $6$ is the largest, followed by $2$.

2. Next is zero: $0$. It is smaller than $6$ and $2$, but larger than any negative integer.

3. The smallest are the negative integers: $-4$ and $-8$. Comparing them, $-4$ is closer to $0$ than $-8$ (or $|-4|=4$ and $|-8|=8$, and $4 < 8$, so $-4 > -8$). Thus, $-4$ is larger than $-8$.

Putting it all together, the integers arranged in descending order are:

$6, 2, 0, -4, -8$

Comparing this with the given options:

(A) $6, 2, 0, -4, -8$ - This matches our result.

(B) $-8, -4, 0, 2, 6$ - This is ascending order.

(C) $0, 2, 6, -4, -8$ - This is not in order (e.g., $0 < 6$).

(D) $6, 0, 2, -4, -8$ - This is not in order (e.g., $0 < 2$).

The correct arrangement in descending order is $6, 2, 0, -4, -8$.

The correct option is (A).

We are asked to find the value of the expression $5 + (-3)$.

This involves adding a positive integer ($5$) and a negative integer ($-3$).

When adding integers with different signs, we can think of it as finding the difference between their absolute values and then applying the sign of the number with the larger absolute value.

Step 1: Find the absolute values of the numbers.

Absolute value of $5$ is $|5| = 5$.

Absolute value of $-3$ is $|-3| = 3$.

Step 2: Subtract the smaller absolute value from the larger absolute value.

The larger absolute value is $5$, and the smaller is $3$.

$5 - 3 = 2$

Step 3: Determine the sign of the result.

The number with the larger absolute value is $5$, which is a positive number. Therefore, the result of the addition will be positive.

So, $5 + (-3) = +2$, which is simply $2$.

Alternatively, adding a negative number is equivalent to subtracting the corresponding positive number.

So, $5 + (-3)$ can be written as $5 - 3$.

$5 - 3 = 2$

The value of $5 + (-3)$ is $2$.

Comparing our result with the given options:

(A) $2$ - This matches our calculated value.

(B) $8$

(C) $-2$

(D) $-8$

The correct value is $2$.

The correct option is (A).

We are asked to calculate the value of $-8 + (-4)$.

This involves adding two negative integers ($-8$ and $-4$).

When adding integers with the same sign, we add their absolute values and keep the common sign.

Step 1: Find the absolute values of the numbers.

Absolute value of $-8$ is $|-8| = 8$.

Absolute value of $-4$ is $|-4| = 4$.

Step 2: Add the absolute values.

$8 + 4 = 12$

Step 3: Keep the common sign.

Both numbers are negative, so the result will be negative.

The common sign is minus ($-$).

So, $-8 + (-4) = -(8+4) = -12$.

Alternatively, adding a negative number is equivalent to subtracting the corresponding positive number.

So, $-8 + (-4)$ can be written as $-8 - 4$.

When subtracting a positive number from a negative number, we move further to the left on the number line. Starting at $-8$, subtracting $4$ means moving $4$ units to the left.

$-8 - 4 = -12$

The value of $-8 + (-4)$ is $-12$.

Comparing our result with the given options:

(A) $-12$ - This matches our calculated value.

(B) $-4$

(C) $4$

(D) $12$

The correct value is $-12$.

The correct option is (A).

The additive inverse of a number is the number that, when added to the original number, results in a sum of zero.

If $a$ is a number, its additive inverse is $-a$, such that $a + (-a) = 0$.

We need to find the additive inverse of the number $9$.

Let the additive inverse of $9$ be $x$. According to the definition, when $9$ and $x$ are added, the result must be $0$.

To find $x$, we can subtract $9$ from both sides of the equation.

Thus, the additive inverse of $9$ is $-9$, because $9 + (-9) = 0$.

Comparing our result with the given options:

(A) $9$: $9 + 9 = 18 \neq 0$.

(B) $-9$: $9 + (-9) = 0$. This is correct.

(C) $0$: $9 + 0 = 9 \neq 0$. $0$ is its own additive inverse ($0+0=0$), but not the additive inverse of $9$.

(D) $\frac{1}{9}$: This is the multiplicative inverse of $9$ ($9 \times \frac{1}{9} = 1$), not the additive inverse.

The additive inverse of $9$ is $-9$.

The correct option is (B).

We are asked to find the value of the expression $12 - (-5)$.

This involves subtracting a negative integer ($-5$) from a positive integer ($12$).

A key rule in integer arithmetic is that subtracting a negative number is the same as adding the corresponding positive number.

That is, for any numbers $a$ and $b$, $a - (-b) = a + b$.

Applying this rule to our expression, where $a = 12$ and $b = 5$ (so $-b = -5$), we have:

Now we perform the addition of the two positive integers:

So, $12 + 5 = 17$.

Therefore, the value of $12 - (-5)$ is $17$.

Comparing our result with the given options:

(A) $7$

(B) $17$ - This matches our calculated value.

(C) $-7$

(D) $-17$

The correct value is $17$.

The correct option is (B).

We are asked to calculate the value of the expression $-10 - 6$.

This involves subtracting a positive integer ($6$) from a negative integer ($-10$).

Subtracting a positive number is the same as adding a negative number.

So, the expression $-10 - 6$ can be rewritten as $-10 + (-6)$.

Now we need to add two negative integers ($-10$ and $-6$).

When adding integers with the same sign (both negative in this case), we add their absolute values and keep the common sign.

Step 1: Find the absolute values of the numbers.

Absolute value of $-10$ is $|-10| = 10$.

Absolute value of $-6$ is $|-6| = 6$.

Step 2: Add the absolute values.

$10 + 6 = 16$

Step 3: Apply the common sign.

Both original numbers are negative, so the sum will be negative.

The result is $-16$.

Therefore, the value of $-10 - 6$ is $-16$.

Comparing our result with the given options:

(A) $4$

(B) $-4$

(C) $16$

(D) $-16$ - This matches our calculated value.

The correct value is $-16$.

The correct option is (D).

We need to simplify the expression $7 + (-2) - (-4)$.

We can simplify this expression by applying the rules for adding and subtracting integers:

• Adding a negative number is equivalent to subtracting the corresponding positive number: $a + (-b) = a - b$.
• Subtracting a negative number is equivalent to adding the corresponding positive number: $a - (-b) = a + b$.

Let's apply these rules to the given expression:

The expression is $7 + (-2) - (-4)$.

First, consider the term $7 + (-2)$. Using the first rule, this is equal to $7 - 2$.

So the expression becomes $5 - (-4)$.

Next, consider the term $5 - (-4)$. Using the second rule, subtracting $-4$ is the same as adding $4$.

Now, we perform the addition:

So, $5 + 4 = 9$.

Therefore, the value of the expression $7 + (-2) - (-4)$ is $9$.

Comparing our result with the given options:

(A) $1$

(B) $5$

(C) $9$ - This matches our calculated value.

(D) $13$

The simplified value is $9$.

The correct option is (C).

Given:

Initial depth of the submarine: 50 metres below sea level.

Ascends: 20 metres.

Descends: 30 metres.

To Find:

The final depth of the submarine relative to sea level.

Solution:

We can represent the sea level as the integer $0$.

Depth below sea level is represented by negative integers.

Distance above sea level (or ascent) is represented by positive integers (added).

Distance below sea level (or descent) is represented by negative integers (added) or by subtracting a positive value.

The initial depth of the submarine is 50 metres below sea level. This can be represented by the integer $-50$.

The submarine ascends 20 metres. This means its position increases by 20 metres. We add $+20$ to the current position.

Position after ascending $=$ Initial position $+$ Ascent

To calculate $-50 + 20$, we find the difference between the absolute values ($|{-50}| = 50$, $|20| = 20$) which is $50 - 20 = 30$. The sign of the result is the sign of the number with the larger absolute value ($-50$), which is negative.

So, after ascending, the submarine is at $-30$ metres relative to sea level (30 metres below sea level).

The submarine then descends 30 metres. This means its position decreases by 30 metres. We add $-30$ to the current position (or subtract $30$).

Final position $=$ Position after ascending $+$ Descent

To calculate $-30 + (-30)$, we add the absolute values ($|-30|=30$, $|-30|=30$) which is $30+30=60$. We keep the common sign, which is negative.

Alternatively, starting at $-30$ and descending 30 metres is $-30 - 30$, which is also $-60$.

The final position of the submarine relative to sea level is $-60$ metres.

A position of $-60$ metres relative to sea level means 60 metres below sea level.

Comparing this result with the given options:

(A) 60 metres below sea level: This corresponds to $-60$.

(B) 60 metres above sea level: This corresponds to $+60$.

(C) 40 metres below sea level: This corresponds to $-40$.

(D) 40 metres above sea level: This corresponds to $+40$.

The calculated final position of $-60$ matches option (A).

The final depth of the submarine is 60 metres below sea level.

The correct option is (A).

We need to examine each statement to determine which one is false.

(A) The sum of two positive integers is always positive.

Let $a$ and $b$ be two positive integers. This means $a > 0$ and $b > 0$.

When we add two positive numbers, the result is always positive.

Example: $3 + 5 = 8$. Both 3 and 5 are positive integers, and their sum, 8, is a positive integer.

This statement is true.

(B) The sum of two negative integers is always negative.

Let $a$ and $b$ be two negative integers. This means $a < 0$ and $b < 0$.

When we add two negative numbers, we add their absolute values and the result is negative.

Example: $-3 + (-5) = -8$. Both -3 and -5 are negative integers, and their sum, -8, is a negative integer.

This statement is true.

(C) The sum of a positive and a negative integer is always positive.

Let $a$ be a positive integer ($a > 0$) and $b$ be a negative integer ($b < 0$).

When adding a positive and a negative integer, the sign of the sum depends on which number has the greater absolute value.

Example 1: $5 + (-3) = 2$. Here, the positive integer (5) has a greater absolute value ($|5|=5$) than the negative integer ($-3$, $|-3|=3$). The sum is positive.

Example 2: $3 + (-5) = -2$. Here, the negative integer ($-5$) has a greater absolute value ($|-5|=5$) than the positive integer ($3$, $|3|=3$). The sum is negative.

Example 3: $5 + (-5) = 0$. Here, the absolute values are equal ($|5|=5$, $|-5|=5$). The sum is zero.

Since the sum can be negative or zero, this statement, which claims the sum is always positive, is false.

(D) The difference between two integers is always an integer.

Let $a$ and $b$ be two integers. Their difference is $a - b$.

The set of integers is closed under the operation of subtraction. This means that when you subtract one integer from another integer, the result is always an integer.

Example: $8 - 5 = 3$. Both 8 and 5 are integers, and 3 is an integer.

Example: $5 - 8 = -3$. Both 5 and 8 are integers, and -3 is an integer.

Example: $-8 - (-5) = -8 + 5 = -3$. Both -8 and -5 are integers, and -3 is an integer.

This statement is true.

Based on the analysis, the statement that is NOT true is (C).

The correct option is (C).

Let's analyze the Assertion and the Reason separately.

Assertion (A): Adding a positive integer to an integer moves you to the right on the number line.

Consider any integer $a$. If we add a positive integer $b$ to $a$, the sum is $a+b$. Since $b$ is positive ($b > 0$), adding $b$ to $a$ results in a number greater than $a$. On a standard horizontal number line, numbers increase as you move to the right. Therefore, starting at $a$ and adding a positive integer $b$ means moving $b$ units in the direction of increasing numbers, which is to the right.

Example: $2 + 3 = 5$. Starting at 2, moving 3 units right lands on 5.

Example: $-5 + 2 = -3$. Starting at -5, moving 2 units right lands on -3.

Thus, Assertion (A) is true.

Reason (R): Adding a negative integer to an integer moves you to the left on the number line.

Consider any integer $a$. If we add a negative integer $c$ to $a$, the sum is $a+c$. Since $c$ is negative ($c < 0$), adding $c$ to $a$ results in a number less than $a$. Adding a negative number is equivalent to subtracting a positive number ($a + (-d) = a - d$, where $d$ is positive). On a standard horizontal number line, numbers decrease as you move to the left. Therefore, starting at $a$ and adding a negative integer $c$ means moving $|c|$ units in the direction of decreasing numbers, which is to the left.

Example: $5 + (-3) = 2$. Starting at 5, adding -3 (or subtracting 3) moves 3 units left, landing on 2.

Example: $-1 + (-2) = -3$. Starting at -1, adding -2 (or subtracting 2) moves 2 units left, landing on -3.

Thus, Reason (R) is true.

Now let's check if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) explains the effect of adding a *positive* integer on movement on the number line. Reason (R) explains the effect of adding a *negative* integer on movement on the number line. While both statements are true facts about integer addition and the number line, Reason (R) does not provide an explanation for why adding a positive integer moves you to the right. It describes a different scenario (adding a negative integer).

Therefore, Reason (R) is a true statement, but it is not the correct explanation for Assertion (A).

Based on our analysis:

• Assertion (A) is true.
• Reason (R) is true.
• Reason (R) is NOT the correct explanation for Assertion (A).

This matches option (B).

The correct option is (B).

We need to match each given situation with its appropriate integer representation based on common conventions.

Let's analyze each situation:

(i) Deposit of $\textsf{₹}$ 2000 in bank: A deposit represents an increase in the amount of money in the bank account. Increases are usually represented by positive integers. Therefore, a deposit of $\textsf{₹}$ 2000 is represented by $+2000$. This matches option (b).

(ii) Withdrawal of $\textsf{₹}$ 500: A withdrawal represents a decrease in the amount of money in the bank account. Decreases are usually represented by negative integers. Therefore, a withdrawal of $\textsf{₹}$ 500 is represented by $-500$. This matches option (a).

(iii) $100^\circ\text{C}$ below freezing point: The freezing point of water is $0^\circ\text{C}$. Temperatures below zero are represented by negative numbers. Therefore, $100^\circ\text{C}$ below freezing point is represented by $-100^\circ\text{C}$. This matches option (c).

(iv) 5 steps to the left from 0: On a standard horizontal number line, the starting point is usually 0. Moving to the left represents decreasing values, and moving 5 steps to the left from 0 means moving to the number that is 5 units less than 0. $0 - 5 = -5$. Therefore, 5 steps to the left from 0 is represented by $-5$. This matches option (d).

Based on the analysis, the correct matching is:

• (i) - (b)
• (ii) - (a)
• (iii) - (c)
• (iv) - (d)

Let's check the given multiple-choice options against our matching:

(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d) - Incorrect.

(B) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d) - Correct.

(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c) - Incorrect.

(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c) - Incorrect.

The correct option is the one that shows the matching (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d).

The correct option is (B).

The additive inverse of an integer is the integer that, when added to the original integer, results in a sum of zero.

Let $a$ be any integer.

Its additive inverse is denoted by $-a$.

According to the definition of additive inverse, the sum of an integer and its additive inverse is always $0$.

Let's consider a few examples:

• For the integer $5$, its additive inverse is $-5$. The sum is $5 + (-5) = 0$.
• For the integer $-10$, its additive inverse is $-(-10) = 10$. The sum is $-10 + 10 = 0$.
• For the integer $0$, its additive inverse is $-0 = 0$. The sum is $0 + 0 = 0$.

In all cases, the sum of an integer and its additive inverse is $0$.

Comparing our finding with the given options:

(A) $1$

(B) $-1$

(C) $0$ - This matches our result.

(D) The integer itself - This is not generally true (e.g., $5 + (-5) = 0$, which is not $5$).

The sum of an integer and its additive inverse is always $0$.

The correct option is (C).

We need to find the value of the expression $-9 + 9$.

This expression involves adding a negative integer ($-9$) and a positive integer ($9$).

Notice that the two numbers, $-9$ and $9$, have the same absolute value ($|-9| = 9$ and $|9| = 9$) but opposite signs. Such numbers are called additive inverses.

The property of additive inverses states that the sum of any integer and its additive inverse is always $0$.

Let the integer be $a$. Its additive inverse is $-a$. Their sum is $a + (-a) = 0$.

In this case, the integer is $9$, and its additive inverse is $-9$. Or, the integer is $-9$, and its additive inverse is $9$.

So, we are calculating the sum of an integer and its additive inverse.

Alternatively, when adding integers with different signs, we subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. Here, the absolute values are equal ($|-9|=9$ and $|9|=9$). The difference is $9 - 9 = 0$. Since $0$ is neither positive nor negative, the result is $0$.

The value of $-9 + 9$ is $0$.

Comparing our result with the given options:

(A) $18$

(B) $-18$

(C) $0$ - This matches our calculated value.

(D) $9$

The correct value is $0$.

The correct option is (C).

We need to evaluate the expression $5 - (-3) - 8$.

We can simplify this expression step by step.

First, let's simplify the subtraction of the negative number: $5 - (-3)$.

Recall that subtracting a negative number is equivalent to adding the corresponding positive number.

Applying this rule to the first part of our expression:

Now, calculate the sum $5 + 3$.

So, the expression $5 - (-3) - 8$ simplifies to $8 - 8$.

Next, calculate the final subtraction $8 - 8$.

Therefore, the value of the expression $5 - (-3) - 8$ is $0$.

Comparing our result with the given options:

(A) $0$ - This matches our calculated value.

(B) $10$

(C) $-10$

(D) $4$

The correct value is $0$.

The correct option is (A).

We are asked to find the integer on the number line that is located 3 units to the left of $-2$.

On a standard horizontal number line:

• Moving to the right corresponds to adding a positive number or moving to a greater value.
• Moving to the left corresponds to subtracting a positive number or moving to a smaller value.

We start at the integer $-2$.

We need to move 3 units to the left. Moving to the left means decreasing the value, which is equivalent to subtracting.

So, we need to subtract 3 from $-2$.

The required integer is $-2 - 3$.

Calculating the subtraction: $-2 - 3$.

Subtracting a positive number is the same as adding the corresponding negative number.

When adding two negative integers, we add their absolute values and keep the negative sign.

$|-2| = 2$ and $|-3| = 3$.

$2 + 3 = 5$.

Keeping the negative sign, the result is $-5$.

Alternatively, we can visualize this on a number line starting from $-2$ and taking 3 steps to the left:

Starting point: $-2$

1 step left: $-2 - 1 = -3$

2 steps left: $-3 - 1 = -4$

3 steps left: $-4 - 1 = -5$

So, 3 units to the left of $-2$ is $-5$.

Comparing our result with the given options:

(A) $1$

(B) $-5$ - This matches our calculated value.

(C) $-1$

(D) $5$

The integer 3 units to the left of $-2$ is $-5$.

The correct option is (B).

Let's define the sets mentioned in the question and the options.

Whole Numbers: The set of whole numbers includes zero and all positive counting numbers (natural numbers). It is represented as $\{0, 1, 2, 3, ...\}$.

Negative Numbers: In the context of discussing integers, negative numbers refer to the negative counting numbers. These are represented as $\{-1, -2, -3, ...\}$.

The question asks about the collection (or union) of whole numbers and negative numbers.

Collection = $\{$ Whole Numbers $\}$ $\cup$ $\{$ Negative Numbers $\}$

Collection = $\{0, 1, 2, 3, ...\} \cup \{-1, -2, -3, ...\}$

Combining these two sets gives us: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Let's check the definitions of the given options:

(A) Natural numbers: Typically, these are the counting numbers $\{1, 2, 3, ...\}$. Some definitions include $0$, but even then, the set does not include negative numbers.

(B) Rational numbers: These are numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include $1, -5, 0, \frac{1}{2}, -\frac{3}{4}, 2.5$. This set includes fractions and decimals, which are not part of the collection of just whole numbers and negative integers.

(C) Integers: The set of integers is defined as the set of all whole numbers and their negatives. It is represented as $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. This precisely matches the collection we formed by combining whole numbers and negative numbers.

(D) Real numbers: This set includes all rational and irrational numbers. Examples include $1, -5, 0, \frac{1}{2}, \sqrt{2}, \pi, 2.5$. This set is much broader than the collection of just whole numbers and negative integers.

The collection of whole numbers and negative numbers forms the set of integers.

The correct option is (C).

We need to find the greatest value among the given options. First, let's evaluate the value of each option.

(A) $|-20|$: The absolute value of $-20$ is its distance from zero on the number line, which is 20. $|-20| = 20$.

(B) $|15|$: The absolute value of $15$ is its distance from zero on the number line, which is 15. $|15| = 15$.

(C) $-25$: This is the integer $-25$.

(D) $10$: This is the integer $10$.

Now we have the values of the options as: $20, 15, -25, 10$.

We need to find the greatest value among these numbers.

Comparing the numbers: $20, 15, -25, 10$.

• Positive numbers are greater than negative numbers and zero. The positive numbers are $20, 15, 10$.
• The negative number is $-25$. It is the smallest value among the options.

Now compare the positive numbers: $20, 15, 10$.

Comparing these, we have $20 > 15 > 10$.

So, the greatest value among $20, 15, 10, -25$ is $20$.

The value $20$ corresponds to option (A) $|-20|$.

The greatest value is $|-20|$.

The correct option is (A).

We are starting at a specific integer on the number line and moving a certain number of steps in a given direction.

The starting position on the number line is $-5$.

We are moving 7 steps to the right.

On a standard horizontal number line:

• Moving to the right corresponds to adding a positive value.
• Moving to the left corresponds to adding a negative value (or subtracting a positive value).

Since we are moving 7 steps to the right, this means we are increasing our position by 7. We can represent this movement as adding $+7$ to our starting position.

The integer we reach is the result of the addition: Starting position $+$ Movement.

To calculate $-5 + 7$, we are adding integers with different signs. We subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.

• Absolute value of $-5$ is $|-5| = 5$.
• Absolute value of $7$ is $|7| = 7$.

The larger absolute value is $7$. The difference between the absolute values is $7 - 5 = 2$.

The number with the larger absolute value is $7$, which is positive. Therefore, the result of the addition is positive.

Alternatively, starting at $-5$ and moving 7 units to the right:

$-5 \xrightarrow{+1} -4 \xrightarrow{+1} -3 \xrightarrow{+1} -2 \xrightarrow{+1} -1 \xrightarrow{+1} 0 \xrightarrow{+1} 1 \xrightarrow{+1} 2$

After 7 steps to the right, we reach the integer $2$.

The integer reached is $2$.

Comparing our result with the given options:

(A) $2$ - This matches our calculated value.

(B) $-12$

(C) $-2$

(D) $12$

The correct option is (A).

Given:

Starting position: House, represented by 0 on a number line.

Movement 1: 10 metres east (positive direction).

Movement 2: 15 metres west (negative direction).

To Find:

The final position relative to the house.

Solution:

We can represent positions and movements on a number line where the house is at 0.

Movement to the east (positive direction) is represented by adding a positive value.

Movement to the west (negative direction) is represented by adding a negative value (or subtracting a positive value).

Initial position $=$ $0$

First movement: 10 metres east. This adds $+10$ to the position.

Position after first movement $=$ Initial position $+$ Movement 1

After walking 10 metres east, the person is at position $10$ (10 metres east of the house).

Second movement: 15 metres west. This adds $-15$ to the current position.

Final position $=$ Position after first movement $+$ Movement 2

To calculate $10 + (-15)$, we are adding integers with different signs. We subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.

• Absolute value of $10$ is $|10| = 10$.
• Absolute value of $-15$ is $|-15| = 15$.

The larger absolute value is $15$. The difference between the absolute values is $15 - 10 = 5$.

The number with the larger absolute value is $-15$, which is negative. Therefore, the result of the addition is negative.

So, the final position is $-5$.

A position of $-5$ relative to the house (at 0) means 5 units in the negative direction. Since the negative direction is west, the final position is 5 metres west of the house.

Comparing this result with the given options:

(A) 5 metres east: This corresponds to $+5$.

(B) 5 metres west: This corresponds to $-5$.

(C) 25 metres east: This corresponds to $+25$.

(D) 25 metres west: This corresponds to $-25$.

The calculated final position of $-5$ matches option (B).

The final position is 5 metres west of their house.

The correct option is (B).

We need to find the sum of the integers $-20$, $10$, and $-5$.

The sum is expressed as: $-20 + 10 + (-5)$.

We can perform the addition step by step, from left to right, or group the positive and negative numbers.

Method 1: Step by step (left to right)

First, calculate $-20 + 10$.

Adding integers with different signs: subtract absolute values ($|{-20}|=20, |10|=10$, difference $= 20-10=10$). Take the sign of the number with the larger absolute value ($-20$, which is negative).

Now, add this result to the next number, $-5$.

$-10 + (-5)$

Adding two negative integers: add absolute values ($|-10|=10, |{-5}|=5$, sum $= 10+5=15$). Keep the common negative sign.

Method 2: Grouping

Group the positive numbers together and the negative numbers together.

Positive numbers: $10$

Negative numbers: $-20, -5$. Their sum is $-20 + (-5) = -25$.

Now add the sum of the positive numbers and the sum of the negative numbers.

Sum $= 10 + (-25)$

Adding integers with different signs: subtract absolute values ($|10|=10, |{-25}|=25$, difference $= 25-10=15$). Take the sign of the number with the larger absolute value ($-25$, which is negative).

Both methods give the same result.

The sum of $-20, 10$, and $-5$ is $-15$.

Comparing our result with the given options:

(A) $35$

(B) $-35$

(C) $-15$ - This matches our calculated value.

(D) $15$

The correct option is (C).

We need to evaluate each given operation and determine if the result is a negative integer.

(A) $5 + (-10)$

Adding a positive and a negative integer. Subtract the absolute values and take the sign of the number with the larger absolute value.

Since $|-10| > |5|$ and $-10$ is negative, the result is negative.

$-5$ is a negative integer.

(B) $-3 - (-8)$

Subtracting a negative number is the same as adding its positive counterpart.

Adding a negative and a positive integer. Subtract the absolute values and take the sign of the number with the larger absolute value.

Since $|8| > |-3|$ and $8$ is positive, the result is positive.

$5$ is a positive integer, not a negative integer.

(C) $-7 + 2$

Adding a negative and a positive integer. Subtract the absolute values and take the sign of the number with the larger absolute value.

Since $|-7| > |2|$ and $-7$ is negative, the result is negative.

$-5$ is a negative integer.

(D) $4 - 9$

Subtracting a larger positive integer from a smaller positive integer. This is equivalent to $4 + (-9)$.

Adding a positive and a negative integer. Subtract the absolute values and take the sign of the number with the larger absolute value.

Since $|-9| > |4|$ and $-9$ is negative, the result is negative.

$-5$ is a negative integer.

The operations that yield a negative integer are (A), (C), and (D).

The correct options are (A), (C), and (D).

The question asks us to complete the sentence regarding the operation of subtracting a negative integer.

Consider the rule for subtracting integers, specifically when subtracting a negative integer. The rule states that subtracting a negative integer is equivalent to adding the corresponding positive integer.

Mathematically, for any two integers $a$ and $b$, the expression $a - (-b)$ is equivalent to $a + b$.

Here, $-b$ is the negative integer, and $b$ is the corresponding positive integer (its absolute value, or its additive inverse). The operation of subtracting $-b$ is replaced by the operation of adding $b$.

Let's look at an example:

Consider $5 - (-3)$. According to the rule, this is equivalent to $5 + 3$.

In this example, subtracting the negative integer $-3$ is the same as adding the corresponding positive integer $3$.

Thus, subtracting a negative integer is equivalent to adding the corresponding positive integer.

Comparing our finding with the given options:

(A) Adding - This matches our conclusion.

(B) Subtracting - This would mean $a - (-b) = a - b$, which is incorrect.

(C) Multiplying - This is not the rule for subtraction.

(D) Dividing - This is not the rule for subtraction.

The correct word to complete the sentence is "Adding".

The correct option is (A).

We need to evaluate the expression $1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10$.

We can group consecutive terms in pairs:

$(1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10)$

Let's evaluate each pair:

$1 - 2 = -1$

$3 - 4 = -1$

$5 - 6 = -1$

$7 - 8 = -1$

$9 - 10 = -1$

Now, substitute these values back into the grouped expression:

$(-1) + (-1) + (-1) + (-1) + (-1)$

Adding these negative integers:

$-1 \times 5 = -5$

So, the value of the expression is $-5$.

Alternate Solution: Step by step calculation

We can evaluate the expression by performing the operations from left to right:

$1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10$

$(1 - 2) + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 = -1 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10$

$(-1 + 3) - 4 + 5 - 6 + 7 - 8 + 9 - 10 = 2 - 4 + 5 - 6 + 7 - 8 + 9 - 10$

$(2 - 4) + 5 - 6 + 7 - 8 + 9 - 10 = -2 + 5 - 6 + 7 - 8 + 9 - 10$

$(-2 + 5) - 6 + 7 - 8 + 9 - 10 = 3 - 6 + 7 - 8 + 9 - 10$

$(3 - 6) + 7 - 8 + 9 - 10 = -3 + 7 - 8 + 9 - 10$

$(-3 + 7) - 8 + 9 - 10 = 4 - 8 + 9 - 10$

$(4 - 8) + 9 - 10 = -4 + 9 - 10$

$(-4 + 9) - 10 = 5 - 10$

$5 - 10 = -5$

Again, the result is $-5$.

The value of the expression is $-5$.

Comparing our result with the given options:

(A) $5$

(B) $-5$ - This matches our calculated value.

(C) $0$

(D) $-10$

The correct option is (B).

Solution:

A number line is a visual representation of numbers on a straight line.

Numbers to the right of $0$ are positive, and numbers to the left of $0$ are negative.

The question states that the tick marks on the number line represent consecutive integers and $0$ is marked.

We are looking for the integer located at point A.

Point A is located to the left of $0$ on the number line.

This means that the integer at point A must be a negative integer.

Let's assume, based on a typical representation for this question and the provided options, that point A is located $3$ tick marks to the left of $0$.

Starting from $0$, moving one tick mark to the left reaches $-1$.

Moving a second tick mark to the left reaches $-2$.

Moving a third tick mark to the left reaches $-3$.

If point A is at the third tick mark to the left of $0$, the integer located at point A is $-3$.

Now let's compare this with the given options:

(A) $-3$

(B) $-4$

(C) $3$

(D) $4$

Our determined value, $-3$, matches option (A).

Therefore, the integer located at point A on the number line is $-3$.

The correct option is (A).

Solution:

Given:

Initial temperature = $5^\circ\text{C}$.

Temperature drop = $7^\circ\text{C}$.

To Find:

The new temperature after the drop.

Solution:

A drop in temperature means the temperature decreases.

Mathematically, a drop is represented by subtraction.

The new temperature will be the initial temperature minus the amount it dropped.

New temperature = Initial temperature - Temperature drop

New temperature $= 5^\circ\text{C} - 7^\circ\text{C}$

We need to calculate $5 - 7$.

When subtracting a larger number from a smaller number, the result is negative.

$5 - 7 = -(7 - 5) = -2$

So, the new temperature is $-2^\circ\text{C}$.

Let's compare this result with the given options:

(A) $2^\circ\text{C}$

(B) $-2^\circ\text{C}$

(C) $12^\circ\text{C}$

(D) $-12^\circ\text{C}$

Our calculated new temperature, $-2^\circ\text{C}$, matches option (B).

Therefore, if the temperature was $5^\circ\text{C}$ and dropped by $7^\circ\text{C}$, the new temperature is $-2^\circ\text{C}$.

The correct option is (B).

Solution:

We are asked to identify the statement that is FALSE about integers.

Let's recall the definitions of the number sets involved:

Natural numbers: The set of positive integers $\{1, 2, 3, ...\}$.

Whole numbers: The set of non-negative integers $\{0, 1, 2, 3, ...\}$.

Integers: The set of whole numbers and their negatives $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

Now let's analyze each statement:

(A) Every natural number is an integer.

The set of natural numbers $\{1, 2, 3, ...\}$ is a subset of the set of integers $\{..., -2, -1, 0, 1, 2, 3, ...\}$. Every natural number is present in the set of integers.

This statement is TRUE.

(B) Every whole number is an integer.

The set of whole numbers $\{0, 1, 2, 3, ...\}$ is a subset of the set of integers $\{..., -2, -1, 0, 1, 2, 3, ...\}$. Every whole number is present in the set of integers.

This statement is TRUE.

(C) There is a largest integer.

The set of integers extends infinitely in the positive direction ($..., 1, 2, 3, ...$). For any integer you choose, say $n$, there is always a larger integer, for example, $n+1$. This means there is no limit to how large an integer can be.

This statement is FALSE.

(D) For every integer, its opposite (additive inverse) is also an integer.

The opposite or additive inverse of an integer $a$ is $-a$. For any integer $a$, its negative, $-a$, is also an integer.

If $a = 5$, its opposite is $-5$, which is an integer.

If $a = -3$, its opposite is $-(-3) = 3$, which is an integer.

If $a = 0$, its opposite is $-0 = 0$, which is an integer.

This statement is TRUE.

We are looking for the FALSE statement. From our analysis, statement (C) is false.

The correct option is (C).

Solution:

We need to evaluate the expression: $15 + (-8) - 3 + (-6)$.

First, let's simplify the signs. Recall that adding a negative number is the same as subtracting that number.

So, $15 + (-8)$ becomes $15 - 8$.

And $3 + (-6)$ becomes $3 - 6$, but in the original expression, it is $-3 + (-6)$, which simplifies to $-3 - 6$.

The expression can be rewritten as:

$15 - 8 - 3 - 6$

Now, we can perform the operations from left to right.

Step 1: Calculate $15 - 8$.

$15 - 8 = 7$

The expression becomes: $7 - 3 - 6$

Step 2: Calculate $7 - 3$.

$7 - 3 = 4$

The expression becomes: $4 - 6$

Step 3: Calculate $4 - 6$.

$4 - 6 = -2$

So, the result of the expression $15 + (-8) - 3 + (-6)$ is $-2$.

Let's compare our result with the given options:

(A) $4$

(B) $-2$

(C) $-4$

(D) $2$

Our calculated value, $-2$, matches option (B).

Therefore, the result is $-2$.

The correct option is (B).

Solution:

An integer is a whole number that can be positive, negative, or zero.

Integers are a subset of real numbers.

They do not include fractions or decimals.

The set of integers, often denoted by $\mathbb{Z}$, can be represented as:

$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

Examples of integers:

Here are three examples of integers:

1. $5$ (a positive integer)

2. $-10$ (a negative integer)

3. $0$ (zero, which is also an integer)

Solution:

To understand how integers extend the set of whole numbers, let's first define both sets.

Whole numbers are the non-negative integers. The set of whole numbers, denoted by $\mathbb{W}$, is:

$\mathbb{W} = \{0, 1, 2, 3, 4, ...\}$

Integers are the set of all whole numbers and their negative counterparts. The set of integers, denoted by $\mathbb{Z}$, is:

$\mathbb{Z} = \{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\}$

Comparing the two sets, we can see that the set of integers contains all the elements of the set of whole numbers.

Additionally, the set of integers includes negative numbers (e.g., $-1, -2, -3, ...$), which are not present in the set of whole numbers.

Therefore, integers extend the set of whole numbers by including the negative integers. The set of integers is essentially the set of whole numbers combined with the set of negative natural numbers.

Inclusion can be shown as:

$\mathbb{W} \subset \mathbb{Z}$

Solution:

We need to represent a "loss of $\textsf{₹}500$" using an integer.

In mathematics, we often use integers to represent quantities that can be positive, negative, or zero.

A loss signifies a decrease in value or amount.

A loss is typically represented by a negative number.

A gain or profit would be represented by a positive number.

The magnitude of the loss is $\textsf{₹}500$.

Since it is a loss, we use the negative sign.

Therefore, a loss of $\textsf{₹}500$ can be represented by the integer $-500$.

The integer representing a loss of $\textsf{₹}500$ is $-500$.

Solution:

We need to represent "10 meters below sea level" using an integer.

Sea level is commonly used as a reference point, often represented by $0$ meters.

Locations above sea level are represented by positive numbers.

Locations below sea level are represented by negative numbers.

The distance below sea level is $10$ meters.

Since the location is below sea level, we use the negative sign.

Therefore, $10$ meters below sea level can be represented by the integer $-10$.

The integer representing $10$ meters below sea level is $-10$.

Solution:

We need to mark the integers $0, -3,$ and $4$ on a number line.

A number line is a line with points marked on it corresponding to numbers.

The number $0$ is typically placed at the center.

Positive integers are placed to the right of $0$, and negative integers are placed to the left of $0$.

The distance between consecutive integers is uniform.

To mark the integers, we follow these steps:

1. Draw a straight line.

2. Mark a point on the line and label it $0$.

3. Mark points at equal intervals to the right of $0$ and label them $1, 2, 3, 4, ...$

4. Mark points at the same equal intervals to the left of $0$ and label them $-1, -2, -3, -4, ...$

Now, we specifically mark the given integers:

The integer $0$ is already marked at the origin.

The integer $-3$ is located $3$ units to the left of $0$.

The integer $4$ is located $4$ units to the right of $0$.

A representation of the number line with the points marked would look like this (text-based representation):

$... -5 \quad -4 \quad \underline{-3} \quad -2 \quad -1 \quad \underline{0} \quad 1 \quad 2 \quad 3 \quad \underline{4} \quad 5 ...$

(Underlined numbers indicate the marked points)

Visually, the number line should show the points corresponding to $0$, $-3$, and $4$ clearly marked.

The number line should extend far enough to include both $-3$ and $4$.

Solution:

We need to determine which of the two integers, $-5$ and $-2$, is greater.

On a number line, numbers increase in value as you move from left to right.

Therefore, the integer that is located further to the right on the number line is the greater integer.

Let's consider the positions of $-5$ and $-2$ on a number line relative to $0$:

$-5$ is located $5$ units to the left of $0$.

$-2$ is located $2$ units to the left of $0$.

Comparing their positions, $-2$ is closer to $0$ (only $2$ units away) than $-5$ (which is $5$ units away).

Since $-2$ is less negative than $-5$, it is located to the right of $-5$ on the number line.

Visually:

$... \quad -6 \quad \underline{-5} \quad -4 \quad -3 \quad \underline{-2} \quad -1 \quad 0 \quad 1 \quad ...$

From this, we can see that $-2$ is to the right of $-5$.

Since $-2$ is to the right of $-5$ on the number line, $-2$ is greater than $-5$.

We can write this inequality as $-2 > -5$ or $-5 < -2$.

Therefore, the greater integer is $-2$.

Solution:

We need to compare the integers $-10$ and $5$ using the symbols $>$ (greater than) or $<$ (less than).

Recall that on a number line, numbers increase as you move from left to right.

The integer $-10$ is a negative integer.

The integer $5$ is a positive integer.

Any positive integer is greater than any negative integer.

On the number line, all negative numbers are located to the left of $0$, and all positive numbers are located to the right of $0$.

This means any positive number is to the right of any negative number.

Since $5$ is a positive integer and $-10$ is a negative integer, $5$ is greater than $-10$.

We can write this relationship using the symbols:

$5 > -10$

Alternatively, we can say that $-10$ is less than $5$.

$-10 < 5$

Both $5 > -10$ and $-10 < 5$ are correct comparisons.

The question asks to use the appropriate symbol ($>$ or $<$) to compare $-10$ and $5$.

Using the symbol $<$ to show that $-10$ is less than $5$ is appropriate.

Using the symbol $>$ to show that $5$ is greater than $-10$ is also appropriate.

Using the first number mentioned in the question ($-10$) as the starting point for the comparison is common practice when presenting the answer with a single inequality.

So, we compare $-10$ relative to $5$.

$-10$ is less than $5$.

The appropriate symbol is $<$.

The comparison is $-10 < 5$.

Solution:

We need to compare the integers $-7$ and $-12$ using the symbols $>$ (greater than) or $<$ (less than).

When comparing negative integers, the integer that is closer to $0$ is the greater integer.

On a number line, numbers increase as you move from left to right.

Let's consider the positions of $-7$ and $-12$ relative to $0$ on a number line:

$-7$ is located $7$ units to the left of $0$.

$-12$ is located $12$ units to the left of $0$.

Since $7 < 12$, $-7$ is closer to $0$ than $-12$.

This means $-7$ is located to the right of $-12$ on the number line.

Visually:

$... \quad -13 \quad \underline{-12} \quad -11 \quad -10 \quad -9 \quad -8 \quad \underline{-7} \quad -6 \quad ...$

Since $-7$ is to the right of $-12$, $-7$ is greater than $-12$.

We can write this relationship using the symbols:

$-7 > -12$

Alternatively, $-12 < -7$.

The question asks to compare $-7$ and $-12$. Starting with $-7$, we state its relationship to $-12$.

$-7$ is greater than $-12$.

The appropriate symbol is $>$.

The comparison is $-7 > -12$.

Solution:

We need to arrange the integers $0, -8, 5, -3,$ and $2$ in ascending order.

Ascending order means arranging the numbers from the smallest to the largest.

Let's list the given integers:

$0, -8, 5, -3, 2$

To arrange integers in ascending order, we consider their positions on the number line. Numbers increase as we move from left to right on the number line.

Negative integers are always smaller than $0$ and positive integers.

Positive integers are always greater than $0$ and negative integers.

Let's identify the negative integers, zero, and positive integers from the list:

Negative integers: $-8, -3$

Zero: $0$

Positive integers: $5, 2$

Now, let's order the negative integers. Among negative integers, the integer with the greater absolute value is smaller. $|-8| = 8$ and $|-3| = 3$. Since $8 > 3$, $-8$ is smaller than $-3$. So, $-8 < -3$.

Ordered negative integers: $-8, -3$.

Now, let's order the positive integers. Among positive integers, the usual comparison applies. $2$ is smaller than $5$. So, $2 < 5$.

Ordered positive integers: $2, 5$.

Combining these with $0$, the order from smallest to largest is:

Negative integers (in ascending order), then $0$, then positive integers (in ascending order).

$-8, -3, 0, 2, 5$

Therefore, the integers in ascending order are: $-8, -3, 0, 2, 5$.

Solution:

We need to find the absolute value of $-9$.

The absolute value of a number is its distance from zero on the number line, regardless of direction.

It is denoted by vertical bars around the number, e.g., $|x|$.

For any real number $x$, the absolute value is defined as:

$|x| = \begin{cases} x & , & \text{if } x \geq 0 \\ -x & , & \text{if } x < 0 \end{cases}$

In this question, the number is $-9$. Since $-9 < 0$, we use the second case of the definition.

$|-9| = -(-9)$

The negative of a negative number is the positive number.

$-(-9) = 9$

Alternatively, we can think of the distance of $-9$ from $0$ on the number line.

To get from $0$ to $-9$, we move $9$ units to the left.

The distance is $9$ units.

Thus, the absolute value of $-9$ is $9$.

$|-9| = 9$

The absolute value of $-9$ is $9$.

Solution:

We need to find the opposite of the integer $15$.

The opposite of an integer is also called its additive inverse.

The opposite of an integer $a$ is the number $-a$, such that when you add the integer and its opposite, the result is $0$.

$a + (-a) = 0$

The given integer is $15$.

We are looking for a number that, when added to $15$, results in $0$.

Let the opposite be $x$. Then:

To find $x$, we can subtract $15$ from both sides of the equation:

$x = 0 - 15$

$x = -15$

So, the opposite of $15$ is $-15$.

We can check this: $15 + (-15) = 15 - 15 = 0$.

Alternatively, by definition, the opposite of an integer $a$ is simply $-a$.

For the integer $15$, the opposite is $-(15) = -15$.

The opposite of the integer $15$ is $-15$.

Solution:

We need to calculate the sum of $8$ and $-5$.

The expression is $8 + (-5)$.

Adding a negative number is the same as subtracting the positive counterpart of that number.

So, $8 + (-5)$ is equivalent to $8 - 5$.

Now, we perform the subtraction:

$8 - 5 = 3$

The result of the calculation is $3$.

Therefore, $8 + (-5) = 3$.

Solution:

We need to calculate the sum of $-12$ and $-7$.

The expression is $-12 + (-7)$.

When adding two negative numbers, we add their absolute values and keep the negative sign.

The absolute value of $-12$ is $|-12| = 12$.

The absolute value of $-7$ is $|-7| = 7$.

Add the absolute values: $12 + 7 = 19$.

Since both numbers are negative, the result is negative.

So, $-12 + (-7) = -(12 + 7) = -19$.

Alternatively, using the rule that adding a negative number is subtraction:

$-12 + (-7) = -12 - 7$

Subtracting a positive number is the same as adding its negative.

Starting at $-12$ on the number line and moving $7$ units further to the left leads to $-19$.

The result of the calculation is $-19$.

Therefore, $-12 + (-7) = -19$.

Solution:

We need to calculate the sum of $-15$ and $20$.

The expression is $-15 + 20$.

This is the sum of a negative integer and a positive integer.

When adding integers with different signs, we find the difference between their absolute values and use the sign of the number with the greater absolute value.

The absolute value of $-15$ is $|-15| = 15$.

The absolute value of $20$ is $|20| = 20$.

The difference between the absolute values is $20 - 15 = 5$.

Now, we determine the sign of the result.

Compare the absolute values: $|20| = 20$ and $|-15| = 15$.

Since $|20| > |-15|$, the sign of the result is the sign of $20$, which is positive.

So, $-15 + 20 = +5 = 5$.

Alternatively, we can think of this as $20 - 15$, which is simpler.

$-15 + 20 = 20 - 15 = 5$

The result of the calculation is $5$.

Therefore, $-15 + 20 = 5$.

Solution:

We need to calculate the difference: $6 - 10$.

This is a subtraction problem where the number being subtracted ($10$) is greater than the number from which it is being subtracted ($6$).

When subtracting a larger number from a smaller number, the result is negative.

We can think of this as finding the difference between $10$ and $6$, and then applying a negative sign because the operation was $6 - 10$ (a smaller number minus a larger number).

Difference between absolute values: $10 - 6 = 4$.

Since we are calculating $6 - 10$, the result is $-4$.

Alternatively, we can think of this on a number line. Starting at $6$, we move $10$ units to the left.

Moving $6$ units to the left from $6$ reaches $0$.

We still need to move another $10 - 6 = 4$ units to the left from $0$.

Moving $4$ units to the left from $0$ reaches $-4$.

The result of the calculation is $-4$.

Therefore, $6 - 10 = -4$.

Solution:

We need to calculate the difference: $5 - (-4)$.

Subtracting a negative number is equivalent to adding the positive version of that number.

The expression $5 - (-4)$ means $5$ minus negative $4$.

This is the same as $5$ plus positive $4$.

So, the expression becomes:

$5 - (-4) = 5 + 4$

Now, perform the addition:

$5 + 4 = 9$

The result of the calculation is $9$.

Therefore, $5 - (-4) = 9$.

Solution:

We need to calculate the difference: $-8 - 3$.

Subtracting a positive number is the same as adding the negative of that number.

So, the expression $-8 - 3$ is equivalent to $-8 + (-3)$.

Now we are adding two negative integers. To add two negative integers, we add their absolute values and place a negative sign in front of the result.

The absolute value of $-8$ is $|-8| = 8$.

The absolute value of $-3$ is $|-3| = 3$.

Add the absolute values: $8 + 3 = 11$.

Place a negative sign in front: $-(11) = -11$.

Alternatively, we can think of this on a number line. Starting at $-8$, we move $3$ units further to the left because we are subtracting $3$.

$-8 - 1 = -9$

$-9 - 1 = -10$

$-10 - 1 = -11$

The result of the calculation is $-11$.

Therefore, $-8 - 3 = -11$.

Solution:

We need to calculate the difference: $-10 - (-6)$.

Subtracting a negative number is the same as adding the positive version of that number.

The expression $-10 - (-6)$ means $-10$ minus negative $6$.

This is the same as $-10$ plus positive $6$.

So, the expression becomes:

$-10 - (-6) = -10 + 6$

Now we are adding integers with different signs (a negative and a positive). We find the difference between their absolute values and use the sign of the number with the greater absolute value.

The absolute value of $-10$ is $|-10| = 10$.

The absolute value of $6$ is $|6| = 6$.

The difference between the absolute values is $10 - 6 = 4$.

Now, we determine the sign of the result.

Compare the absolute values: $|-10| = 10$ and $|6| = 6$.

Since $|-10| > |6|$, the sign of the result is the sign of $-10$, which is negative.

So, $-10 + 6 = -4$.

Alternatively, we can think of this on a number line. Starting at $-10$, we move $6$ units to the right because we are adding $6$.

Moving $6$ units to the right from $-10$ reaches $-4$.

The result of the calculation is $-4$.

Therefore, $-10 - (-6) = -4$.

Solution:

We need to find the value of the expression: $7 + (-3) + 5$.

We can simplify the expression by performing the operations from left to right.

Recall that adding a negative number is the same as subtracting.

So, $7 + (-3)$ is the same as $7 - 3$.

Step 1: Calculate $7 + (-3)$.

$7 + (-3) = 7 - 3 = 4$

The expression becomes: $4 + 5$

Step 2: Calculate $4 + 5$.

$4 + 5 = 9$

Alternatively, we can group the positive numbers and negative numbers.

Positive numbers: $7, 5$. Their sum is $7 + 5 = 12$.

Negative numbers: $-3$.

The expression is the sum of $12$ and $-3$.

$12 + (-3) = 12 - 3 = 9$.

The value of the expression is $9$.

Therefore, $7 + (-3) + 5 = 9$.

Solution:

We need to find the value of the expression: $-2 - (-4) + (-1)$.

Let's simplify the signs in the expression first.

Recall that:

Subtracting a negative number is the same as adding the positive number: $-(-4) = +4$.

Adding a negative number is the same as subtracting the positive number: $+(-1) = -1$.

The expression becomes:

$-2 + 4 - 1$

Now, we perform the operations from left to right.

Step 1: Calculate $-2 + 4$.

$-2 + 4 = 2$ (This is the same as $4 - 2$).

The expression becomes: $2 - 1$

Step 2: Calculate $2 - 1$.

$2 - 1 = 1$

Alternatively, we can group the positive and negative numbers.

Positive numbers: $+4$.

Negative numbers: $-2, -1$. Their sum is $-2 + (-1) = -3$.

The expression is the sum of $-3$ and $+4$.

$-3 + 4 = 4 - 3 = 1$.

The value of the expression is $1$.

Therefore, $-2 - (-4) + (-1) = 1$.

Solution:

We need to find the integer that is $4$ less than $-1$.

The phrase "$4$ less than $-1$" translates mathematically to starting with $-1$ and subtracting $4$ from it.

The mathematical expression is: $-1 - 4$.

To calculate $-1 - 4$, we are essentially starting at $-1$ on the number line and moving $4$ units to the left.

Alternatively, we can think of this as the sum of $-1$ and $-4$ because subtracting a positive number is the same as adding its negative.

$-1 - 4 = -1 + (-4)$

When adding two negative numbers, we add their absolute values and use a negative sign.

The absolute value of $-1$ is $|-1| = 1$.

The absolute value of $-4$ is $|-4| = 4$.

Sum of absolute values: $1 + 4 = 5$.

Apply the negative sign: $-5$.

Thus, $-1 - 4 = -5$.

The integer that is $4$ less than $-1$ is $-5$.

Solution:

We need to find the integer that is $3$ more than $-5$.

The phrase "$3$ more than $-5$" means we start with the integer $-5$ and add $3$ to it.

The mathematical expression for this is: $-5 + 3$.

To calculate the sum of a negative integer ($-5$) and a positive integer ($3$), we follow these steps:

1. Find the absolute values of the two numbers: $|-5| = 5$ and $|3| = 3$.

2. Find the difference between the absolute values: $5 - 3 = 2$.

3. The result takes the sign of the number with the greater absolute value. Since $|-5| = 5$ is greater than $|3| = 3$, the result will have the same sign as $-5$, which is negative.

So, $-5 + 3 = -2$.

Alternatively, think of a number line. Start at $-5$. Moving "$3$ more" means moving $3$ units to the right.

Starting at $-5$, move 1 unit right to $-4$, 1 unit right to $-3$, and 1 unit right to $-2$.

The final position is $-2$.

The integer that is $3$ more than $-5$ is $-2$.

Solution:

Given:

Initial temperature of the city = $5^\circ\text{C}$.

Temperature drop = $7^\circ\text{C}$.

To Find:

The new temperature after the drop.

Solution:

When the temperature "drops", it means the temperature decreases. Mathematically, a drop is represented by subtraction.

To find the new temperature, we subtract the amount of the drop from the initial temperature.

New temperature = Initial temperature - Temperature drop

New temperature = $5^\circ\text{C} - 7^\circ\text{C}$

We need to calculate $5 - 7$.

When subtracting a larger number from a smaller number, the result is negative.

$5 - 7 = -(7 - 5) = -2$

So, the new temperature is $-2^\circ\text{C}$.

The new temperature of the city is $-2^\circ\text{C}$.

Solution:

Given:

Initial bank balance = $\textsf{₹}1000$.

Withdrawal amount = $\textsf{₹}1200$.

To Find:

The new balance in the bank account.

Solution:

A withdrawal means money is taken out of the account, which decreases the balance.

To find the new balance, we subtract the withdrawal amount from the initial balance.

New balance = Initial balance - Withdrawal amount

New balance $= \textsf{₹}1000 - \textsf{₹}1200$

We need to calculate $1000 - 1200$.

This is a subtraction where the number being subtracted ($1200$) is larger than the number from which it is being subtracted ($1000$). The result will be negative.

$1000 - 1200 = -(1200 - 1000) = -200$

So, the new balance is $-\textsf{₹}200$.

A negative balance in a bank account represents an overdraft or debt.

The new balance in the bank account is $-\textsf{₹}200$.

Solution:

Given:

A number line segment is shown.

The first marked point on the segment represents the integer $-5$.

The subsequent marked points represent consecutive integers moving to the right from $-5$.

Point A is located at one of these marked points.

To Find:

The integer represented by point A.

Solution:

The number line starts with the integer $-5$.

Since the marked points to the right are consecutive integers, they will follow in increasing order, with a difference of $1$ between each consecutive pair.

Starting from $-5$, the sequence of consecutive integers to the right is:

First point: $-5$

Second point: $-5 + 1 = -4$

Third point: $-4 + 1 = -3$

Fourth point: $-3 + 1 = -2$

Fifth point: $-2 + 1 = -1$

And so on: $-1 + 1 = 0$, $0 + 1 = 1$, $1 + 1 = 2$, etc.

The integers represented by the marked points on the segment, starting from the left, are $-5, -4, -3, -2, -1, 0, 1, 2, ...$

Point A is located at one specific marked point on this segment as shown in the image.

However, the image accompanying the question is not available here.

Without the image, it is impossible to determine the exact location of point A among the marked consecutive integers starting from $-5$.

For example:

If Point A is the second marked point from the left, it represents $-4$.

If Point A is the third marked point from the left, it represents $-3$.

If Point A is the fourth marked point from the left, it represents $-2$.

And so on.

Therefore, the integer represented by point A can only be identified by observing its position on the provided number line image.

Based on the typical representation in such problems, point A is likely marked at a specific integer value among $-4, -3, -2, -1, 0, 1, 2, ...$ on the segment that starts at $-5$.

Solution:

Concept of Integers using a Number Line:

A number line is a straight line on which points are marked to correspond to numbers. It is a fundamental tool for visualizing numbers and their relationships.

Integers are the set of whole numbers ($\{0, 1, 2, 3, ...\}$) and their opposites (negative natural numbers $\{..., -3, -2, -1\}$). The set of integers is denoted by $\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

On a number line, integers are represented by equally spaced points. The number $0$ is usually placed at a central point called the origin. Positive integers are placed to the right of $0$, and negative integers are placed to the left of $0$.

Representation of Integers on a Number Line:

We need to represent the integers $-6, -1, 0, 3, 7$ on a number line.

1. Draw a straight line and mark a point roughly in the middle as the origin, labeling it $0$.

2. Mark points to the right of $0$ at equal intervals and label them $1, 2, 3, 4, 5, 6, 7, ...$

3. Mark points to the left of $0$ at the same equal intervals and label them $-1, -2, -3, -4, -5, -6, -7, ...$

4. Now, locate the given integers: $0, -6, -1, 3, 7$ on this number line and mark them clearly.

A visual representation (text-based approximation):

$... \quad \underline{-6} \quad -5 \quad -4 \quad -3 \quad -2 \quad \underline{-1} \quad \underline{0} \quad 1 \quad 2 \quad \underline{3} \quad 4 \quad 5 \quad 6 \quad \underline{7} \quad ...$

(The underlined numbers indicate the positions of the integers $-6, -1, 0, 3,$ and $7$)

Significance of Positive and Negative Signs and the Position of Zero:

The number line helps in understanding the significance of positive and negative signs and the role of zero:

1. Position of Zero ($0$): Zero is the origin or reference point on the number line. It separates the positive numbers from the negative numbers. Zero is neither positive nor negative.

2. Positive Numbers (with '+' sign or no sign): Numbers located to the right of zero are positive integers ($1, 2, 3, ...$). They represent values greater than zero. Positive signs are often omitted but understood.

3. Negative Numbers (with '-' sign): Numbers located to the left of zero are negative integers ($..., -3, -2, -1$). They represent values less than zero. The negative sign indicates direction and value relative to zero.

Relationship between numbers on the Number Line:

On the number line, the value of the numbers increases as you move from left to right and decreases as you move from right to left.

For example, $3$ is to the right of $-1$, so $3 > -1$. Similarly, $-6$ is to the left of $-1$, so $-6 < -1$. This positional relationship visually demonstrates the concept of greater than ($>$) and less than ($<$) for integers.

In summary, the number line provides a clear visual model where position indicates value and the sign (positive or negative) indicates direction relative to the central point, zero.

Solution:

Comparing Integers:

To compare two integers, we can visualize their positions on a number line.

On a number line, numbers increase in value as you move from left to right.

Therefore, if an integer $a$ is to the right of an integer $b$ on the number line, then $a$ is greater than $b$ ($a > b$). Conversely, if an integer $a$ is to the left of an integer $b$, then $a$ is less than $b$ ($a < b$).

Let's apply this concept to the given pairs of integers:

(a) Compare $-10$ and $-4$.

Consider a number line:

$... \quad -11 \quad \underline{-10} \quad -9 \quad -8 \quad -7 \quad -6 \quad -5 \quad \underline{-4} \quad -3 \quad ...$

The integer $-10$ is located $10$ units to the left of $0$.

The integer $-4$ is located $4$ units to the left of $0$.

Comparing their positions, $-4$ is to the right of $-10$ on the number line.

Therefore, $-4$ is greater than $-10$.

Using the symbol, we can write: $-10 < -4$ or $-4 > -10$.

Typically, we start the comparison with the first number given.

Comparison: $-10 < -4$.

(b) Compare $0$ and $-6$.

Consider a number line:

$... \quad -7 \quad \underline{-6} \quad -5 \quad -4 \quad -3 \quad -2 \quad -1 \quad \underline{0} \quad 1 \quad 2 \quad ...$

The integer $0$ is the origin.

The integer $-6$ is located $6$ units to the left of $0$.

Comparing their positions, $0$ is to the right of $-6$ on the number line.

Therefore, $0$ is greater than $-6$.

Using the symbol, we can write: $0 > -6$ or $-6 < 0$.

Comparison: $0 > -6$.

(c) Compare $8$ and $-9$.

Consider a number line (extended to include these values):

$... \quad -10 \quad \underline{-9} \quad -8 \quad ... \quad 0 \quad ... \quad 7 \quad \underline{8} \quad 9 \quad ...$

The integer $8$ is a positive integer, located $8$ units to the right of $0$.

The integer $-9$ is a negative integer, located $9$ units to the left of $0$.

Comparing their positions, $8$ is to the right of $-9$ on the number line (all positive numbers are to the right of all negative numbers).

Therefore, $8$ is greater than $-9$.

Using the symbol, we can write: $8 > -9$ or $-9 < 8$.

Comparison: $8 > -9$.

Solution:

Concept of Absolute Value:

The absolute value of an integer represents its distance from zero on the number line. Distance is always a non-negative quantity.

The absolute value of an integer $x$ is denoted by $|x|$.

Formally, the absolute value of $x$ is defined as:

$|x| = \begin{cases} x & , & \text{if } x \geq 0 \\ -x & , & \text{if } x < 0 \end{cases}$

This means that if the integer is positive or zero, its absolute value is the integer itself. If the integer is negative, its absolute value is the opposite of the integer (which is positive).

Absolute value for given integers:

For $25$: Since $25 \geq 0$, $|25| = 25$.

For $-30$: Since $-30 < 0$, $|-30| = -(-30) = 30$.

For $0$: Since $0 \geq 0$, $|0| = 0$.

Concept of the Opposite of an Integer (Additive Inverse):

The opposite of an integer is the integer that is the same distance from zero on the number line but on the opposite side.

The opposite of an integer $a$ is denoted by $-a$. When an integer is added to its opposite, the result is zero ($a + (-a) = 0$). This is why the opposite is also called the additive inverse.

Opposite for given integers:

For $25$: The opposite of $25$ is $-25$, because $25 + (-25) = 0$.

For $-30$: The opposite of $-30$ is $-(-30) = 30$, because $-30 + 30 = 0$.

For $0$: The opposite of $0$ is $-0 = 0$, because $0 + 0 = 0$.

Can the absolute value of an integer be negative?

No, the absolute value of an integer cannot be negative.

Explanation:

The absolute value represents a distance from zero on the number line. Distance is a measure of how far apart two points are, and it is always a non-negative value (zero or positive).

Consider the definition $|x| = \begin{cases} x & , & \text{if } x \geq 0 \\ -x & , & \text{if } x < 0 \end{cases}$.

If $x$ is a non-negative integer ($x \geq 0$), then $|x| = x$, which is non-negative.

If $x$ is a negative integer ($x < 0$), then $|x| = -x$. Since $x$ is negative, $-x$ is positive. For example, if $x = -5$, then $|x| = -(-5) = 5$, which is positive.

In both cases ($x \geq 0$ and $x < 0$), the result of the absolute value operation $|x|$ is always a non-negative number ($|x| \geq 0$).

Therefore, the absolute value of any integer is always zero or a positive number.

Solution:

Rules for Adding Integers:

Rule 1: Adding Integers with the Same Sign

To add two integers with the same sign (both positive or both negative):

1. Add their absolute values.

2. The sum has the same sign as the original integers.

Examples (Same Sign):

Example 1 (Both Positive): $4 + 7$

Absolute values are $|4| = 4$ and $|7| = 7$.

Add the absolute values: $4 + 7 = 11$.

Since both original numbers were positive, the sum is positive.

$4 + 7 = 11$

Example 2 (Both Negative): $-5 + (-8)$

Absolute values are $|-5| = 5$ and $|-8| = 8$.

Add the absolute values: $5 + 8 = 13$.

Since both original numbers were negative, the sum is negative.

$-5 + (-8) = -13$

Rule 2: Adding Integers with Different Signs

To add two integers with different signs (one positive and one negative):

1. Find the difference between their absolute values (subtract the smaller absolute value from the larger absolute value).

2. The sum has the same sign as the integer with the greater absolute value.

Examples (Different Signs):

Example 1 (Positive + Negative): $10 + (-3)$

Absolute values are $|10| = 10$ and $|-3| = 3$.

Difference between absolute values: $10 - 3 = 7$.

Compare absolute values: $|10| = 10 > |-3| = 3$. The sign of the result is the sign of $10$, which is positive.

$10 + (-3) = 7$

Example 2 (Negative + Positive): $-15 + 6$

Absolute values are $|-15| = 15$ and $|6| = 6.

Difference between absolute values: $15 - 6 = 9$.

Compare absolute values: $|-15| = 15 > |6| = 6$. The sign of the result is the sign of $-15$, which is negative.

$-15 + 6 = -9$

Solution:

We will use a number line to demonstrate the addition of integers.

(a) Show the addition of $5 + (-3)$ on a number line.

Explanation:

1. Start at the position of the first integer, which is $5$. Locate the point $5$ on the number line.

2. The second integer is $-3$. Since we are adding a negative integer, we move to the left on the number line.

3. The value of the second integer is $3$ (absolute value is $|-3|=3$), so we move $3$ units to the left from our starting point, $5$.

Starting at $5$, move 1 unit left to $4$.

Move another 1 unit left from $4$ to $3$.

Move a third unit left from $3$ to $2$.

4. The point where we stop is the sum. We stopped at $2$.

Number line representation (text-based approximation):

$... \quad 0 \quad 1 \quad \underline{2} \quad 3 \quad 4 \quad \underline{5} \quad 6 \quad ...$

(Starting at 5, move 3 units left to reach 2)

Therefore, $5 + (-3) = 2$.

(b) Show the addition of $-4 + (-2)$ on a number line.

Explanation:

1. Start at the position of the first integer, which is $-4$. Locate the point $-4$ on the number line.

2. The second integer is $-2$. Since we are adding a negative integer, we move to the left on the number line.

3. The value of the second integer is $2$ (absolute value is $|-2|=2$), so we move $2$ units to the left from our starting point, $-4$.

Starting at $-4$, move 1 unit left to $-5$.

Move another 1 unit left from $-5$ to $-6$.

4. The point where we stop is the sum. We stopped at $-6$.

Number line representation (text-based approximation):

$... \quad \underline{-6} \quad -5 \quad \underline{-4} \quad -3 \quad ... \quad 0 \quad ...$

(Starting at -4, move 2 units left to reach -6)

Therefore, $-4 + (-2) = -6$.

Solution:

Concept of Subtraction as Addition of Additive Inverse:

Subtraction of integers can be defined in terms of addition. Subtracting an integer is the same as adding its additive inverse (or opposite).

The additive inverse of an integer $a$ is the integer $-a$ such that $a + (-a) = 0$.

The rule for subtraction is: For any two integers $a$ and $b$, $a - b = a + (-b)$.

In other words, to subtract an integer, change the subtraction sign to an addition sign and change the sign of the integer being subtracted to its opposite.

Let's use this concept to calculate the given expressions:

(a) Calculate $10 - 18$.

Using the concept of additive inverse, subtracting $18$ is the same as adding the additive inverse of $18$, which is $-18$.

$10 - 18 = 10 + (-18)$

Now we add integers with different signs. The absolute values are $|10| = 10$ and $|-18| = 18$. The difference between the absolute values is $18 - 10 = 8$. The number with the greater absolute value is $-18$, which is negative. So the result is negative.

$10 + (-18) = -8$

(b) Calculate $-7 - 5$.

Using the concept of additive inverse, subtracting $5$ is the same as adding the additive inverse of $5$, which is $-5$.

$-7 - 5 = -7 + (-5)$

Now we add integers with the same sign (both negative). Add their absolute values: $|-7| = 7$ and $|-5| = 5$. $7 + 5 = 12$. Since both numbers are negative, the sum is negative.

$-7 + (-5) = -12$

(c) Calculate $12 - (-6)$.

Using the concept of additive inverse, subtracting $-6$ is the same as adding the additive inverse of $-6$. The additive inverse of $-6$ is $-(-6) = 6$.

$12 - (-6) = 12 + 6$

Now we add two positive integers.

$12 + 6 = 18$

(d) Calculate $-15 - (-8)$.

Using the concept of additive inverse, subtracting $-8$ is the same as adding the additive inverse of $-8$. The additive inverse of $-8$ is $-(-8) = 8$.

$-15 - (-8) = -15 + 8$

Now we add integers with different signs. The absolute values are $|-15| = 15$ and $|8| = 8$. The difference between the absolute values is $15 - 8 = 7$. The number with the greater absolute value is $-15$, which is negative. So the result is negative.

$-15 + 8 = -7$

Solution:

We will use a number line to demonstrate the subtraction of integers. We will use the concept that $a - b = a + (-b)$.

(a) Show the subtraction of $2 - 7$ on a number line.

The expression is $2 - 7$. Using the additive inverse concept, this is equivalent to $2 + (-7)$.

Explanation:

1. Start at the position of the first integer, which is $2$. Locate the point $2$ on the number line.

2. We are subtracting $7$, which is equivalent to adding $-7$. Since we are adding a negative integer, we move to the left on the number line.

3. We move $7$ units to the left from our starting point, $2$.

Starting at $2$, move 1 unit left to $1$.

Move 1 unit left from $1$ to $0$. (Total 2 units moved)

Move 1 unit left from $0$ to $-1$. (Total 3 units moved)

Move 1 unit left from $-1$ to $-2$. (Total 4 units moved)

Move 1 unit left from $-2$ to $-3$. (Total 5 units moved)

Move 1 unit left from $-3$ to $-4$. (Total 6 units moved)

Move 1 unit left from $-4$ to $-5$. (Total 7 units moved)

4. The point where we stop is the result of the subtraction. We stopped at $-5$.

Number line representation (text-based approximation):

$... \quad -6 \quad \underline{-5} \quad -4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad \underline{2} \quad 3 \quad ...$

(Starting at 2, move 7 units left to reach -5)

Therefore, $2 - 7 = -5$.

(b) Show the subtraction of $-3 - (-5)$ on a number line.

The expression is $-3 - (-5)$. Using the additive inverse concept, subtracting $-5$ is the same as adding the opposite of $-5$, which is $5$. So, the expression is equivalent to $-3 + 5$.

Explanation:

1. Start at the position of the first integer, which is $-3$. Locate the point $-3$ on the number line.

2. We are adding $5$. Since we are adding a positive integer, we move to the right on the number line.

3. We move $5$ units to the right from our starting point, $-3$.

Starting at $-3$, move 1 unit right to $-2$.

Move 1 unit right from $-2$ to $-1$.

Move 1 unit right from $-1$ to $0$.

Move 1 unit right from $0$ to $1$.

Move 1 unit right from $1$ to $2$.

4. The point where we stop is the result of the subtraction. We stopped at $2$.

Number line representation (text-based approximation):

$... \quad -4 \quad \underline{-3} \quad -2 \quad -1 \quad 0 \quad 1 \quad \underline{2} \quad 3 \quad ...$

(Starting at -3, move 5 units right to reach 2)

Therefore, $-3 - (-5) = 2$.

Solution:

We need to simplify the expression: $20 + (-15) - (-10) + (-8) - 2$.

We will simplify the signs and perform the operations step-by-step.

First, let's rewrite the expression by handling the signs:

Recall that:

Adding a negative number is equivalent to subtracting: $+(-\text{a}) = -\text{a}$.

Subtracting a negative number is equivalent to adding the positive number: $- (-\text{a}) = +\text{a}$.

Applying these rules:

$20 + (-15)$ becomes $20 - 15$.

$- (-10)$ becomes $+ 10$.

$+ (-8)$ becomes $- 8$.

$- 2$ remains $- 2$ (or $+ (-2)$).

So, the expression can be rewritten as:

$20 - 15 + 10 - 8 - 2$

Now, perform the operations from left to right:

Step 1: Calculate $20 - 15$.

$20 - 15 = 5$

The expression becomes: $5 + 10 - 8 - 2$

Step 2: Calculate $5 + 10$.

$5 + 10 = 15$

The expression becomes: $15 - 8 - 2$

Step 3: Calculate $15 - 8$.

$15 - 8 = 7$

The expression becomes: $7 - 2$

Step 4: Calculate $7 - 2$.

$7 - 2 = 5$

Alternatively, we could group the positive and negative numbers:

Positive numbers: $20, 10$. Sum = $20 + 10 = 30$.

Negative numbers: $-15, -8, -2$. Sum = $(-15) + (-8) + (-2) = -(15 + 8 + 2) = -(23 + 2) = -25$.

Now, add the sum of positive numbers and the sum of negative numbers:

$30 + (-25) = 30 - 25 = 5$.

The value of the expression is $5$.

Therefore, $20 + (-15) - (-10) + (-8) - 2 = 5$.

Solution:

Given:

Initial altitude = $1500$ meters above sea level.

Descent = $800$ meters.

Ascent = $450$ meters.

Representation: Altitude above sea level is positive, descent is negative (subtraction), ascent is positive (addition).

To Find:

The mountaineer's final altitude.

Solution:

We start with the initial altitude, which is $1500$ meters above sea level. This is represented by the integer $+1500$ or simply $1500$.

He descends $800$ meters. A descent means a decrease in altitude. We can represent this as subtracting $800$ or adding $-800$. Let's use subtraction directly:

Altitude after descent = Initial altitude - Descent

Altitude after descent $= 1500 - 800$

Calculating the first step:

$1500 - 800 = 700$

So, after descending, his altitude is $700$ meters above sea level.

Next, he ascends $450$ meters. An ascent means an increase in altitude. We represent this as adding $450$.

Final altitude = Altitude after descent + Ascent

Final altitude $= 700 + 450$

Calculating the second step:

$700 + 450 = 1150$

So, his final altitude is $1150$ meters above sea level.

We can write the entire process as a single expression:

Final altitude $= 1500 - 800 + 450$

Final altitude $= (1500 - 800) + 450$

Final altitude $= 700 + 450$

Final altitude $= 1150$

The final altitude of the mountaineer is $1150$ meters above sea level.

Solution:

Given:

Marks for a correct answer = $+5$

Marks for an incorrect answer = $-2$

Number of correct answers by Rahul = $7$

Number of incorrect answers by Rahul = $3$

To Find:

Rahul's total score in the quiz.

Solution:

First, we calculate the total marks obtained for correct answers.

Marks for correct answers = (Number of correct answers) $\times$ (Marks per correct answer)

Marks for correct answers $= 7 \times 5 = 35$

Next, we calculate the total marks obtained for incorrect answers.

Marks for incorrect answers = (Number of incorrect answers) $\times$ (Marks per incorrect answer)

Marks for incorrect answers $= 3 \times (-2)$

When multiplying a positive integer by a negative integer, the result is a negative integer.

$3 \times (-2) = -6$

Now, we find the total score by adding the marks for correct answers and the marks for incorrect answers.

Total score = Marks for correct answers + Marks for incorrect answers

Total score $= 35 + (-6)$

Adding a negative integer is the same as subtracting the corresponding positive integer.

$35 + (-6) = 35 - 6$

Performing the subtraction:

$35 - 6 = 29$

So, Rahul's total score is $29$.

Rahul's total score is $29$.

Solution:

Given:

Initial temperature = $-3^\circ\text{C}$.

Temperature rise during the day = $10^\circ\text{C}$.

Temperature fall during the night = $4^\circ\text{C}$.

To Find:

The temperature at the end of the night.

Solution:

We start with the initial temperature.

Initial temperature = $-3^\circ\text{C}$.

During the day, the temperature rose by $10^\circ\text{C}$. A rise in temperature means an increase, which is represented by addition.

Temperature at the end of the day = Initial temperature + Temperature rise

Temperature at the end of the day $= -3^\circ\text{C} + 10^\circ\text{C}$

Calculate $-3 + 10$. This is addition of integers with different signs. $|-3|=3$, $|10|=10$. Difference $= 10-3=7$. Sign is positive because $|10| > |-3|$.

Temperature at the end of the day $= 7^\circ\text{C}$.

At night, the temperature fell by $4^\circ\text{C}$. A fall in temperature means a decrease, which is represented by subtraction.

Temperature at the end of the night = Temperature at the end of the day - Temperature fall

Temperature at the end of the night $= 7^\circ\text{C} - 4^\circ\text{C}$

Calculate $7 - 4$.

$7 - 4 = 3$

Temperature at the end of the night $= 3^\circ\text{C}$.

We can also represent the entire process as a single expression:

Final temperature $= -3 + 10 - 4$

Final temperature $= (-3 + 10) - 4$

Final temperature $= 7 - 4$

Final temperature $= 3$

The temperature at the end of the night was $3^\circ\text{C}$.

Solution:

We can represent the ground floor as $0$.

Floors above the ground level can be represented by positive integers.

Floors below the ground level (basement floors) can be represented by negative integers.

Given:

Initial position = Ground floor ($0$).

Movement 1: Goes down to the basement, $3$ floors below ground level. This corresponds to moving from $0$ down by $3$ floors, reaching floor $-3$. Mathematically, this is $0 - 3$ or $0 + (-3)$.

Movement 2: Goes up $10$ floors from the current position (the basement floor).

To Find:

The final floor level of the lift.

Solution:

Step 1: Initial position is $0$.

Step 2: The lift goes down $3$ floors. This means the position changes by $-3$.

Position after going down $= 0 - 3 = -3$.

The lift is now on the 3rd basement floor, which is floor $-3$.

Step 3: From floor $-3$, the lift goes up $10$ floors. Going up means adding the number of floors moved upwards.

Final position = Position after going down + Floors moved up

Final position $= -3 + 10$

Calculate $-3 + 10$. This is addition of integers with different signs. $|-3|=3$, $|10|=10$. Difference $= 10-3=7$. Sign is positive because $|10| > |-3|$.

Final position $= 7$.

The final position of the lift is the 7th floor above ground level.

The lift is now on the 7th floor.