Chapter 6 Integers

Given:

We need to consider the integers on a number line.

The range specified is between $-8$ and $-2$.

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To Find:

1. The integers that lie strictly between $-8$ and $-2$.

2. The largest integer among those found.

3. The smallest integer among those found.

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Solution:

To find the integers between $-8$ and $-2$, we look for integers $x$ such that $-8 < x < -2$.

Visualizing the number line, we start just after $-8$ and stop just before $-2$.

Moving from left to right (increasing value) from $-8$, the integers are:

..., $-9$, $-8$, $-7$, $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$, $1$, ...

The integers that are greater than $-8$ and less than $-2$ are:

$-7, -6, -5, -4, -3$

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Now, we need to find the largest and smallest integers among $-7, -6, -5, -4, -3$.

On the number line, the integer furthest to the right is the largest, and the integer furthest to the left is the smallest.

Comparing the integers $-7, -6, -5, -4, -3$:

The largest integer is $\mathbf{-3}$.

The smallest integer is $\mathbf{-7}$.

Solution (a):

The starting position of the button is at $-3$.

The target position is at $-9$.

To determine the direction, we compare the target position ($-9$) with the starting position ($-3$). Since $-9$ is a smaller number than $-3$, we need to move towards the left on the number line.

To find the number of steps, we can count the distance between the two points, as shown on the number line below.

The number of steps = $|-9 - (-3)| = |-9 + 3| = |-6| = 6$.

Therefore, we should move 6 steps to the left.

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Solution (b):

The starting position is at $-6$.

We need to move 4 steps to the right.

Moving to the right on the number line means adding a positive value to the starting number. This movement is illustrated on the number line below.

Final Position = Starting Position + Number of Steps to the Right

Final Position = $-6 + 4$

Final Position = $-2$.

Therefore, if we move 4 steps to the right of $-6$, we will reach the number $\mathbf{-2}$.

(a) The opposite of 'Increase in weight' is Decrease in weight.

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(b) The opposite of '30 km north' is 30 km south.

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(c) The opposite of '80 m east' is 80 m west.

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(d) The opposite of 'Loss of $\textsf{₹}$ 700' is Gain of $\textsf{₹}$ 700 or Profit of $\textsf{₹}$ 700.

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(e) The opposite of '100 m above sea level' is 100 m below sea level.

(a) An aeroplane flying above the ground indicates a positive height.

Height = 2000 metres.

Integer representation: +2000 m

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(b) A submarine moving below the sea level indicates a negative depth.

Depth = 800 metres.

Integer representation: -800 m

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(c) A deposit represents an increase in amount, indicated by a positive sign.

Amount = Rupees 200.

Integer representation: +$\textsf{₹}$ 200

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(d) A withdrawal represents a decrease in amount, indicated by a negative sign.

Amount = Rupees 700.

Integer representation: -$\textsf{₹}$ 700

To represent these numbers on a number line, we draw a horizontal line with zero ($0$) at the center. We can assign a letter to represent the point for each number. Positive integers are to the right of zero, and negative integers are to the left.

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(a) + 5

Let the point representing +5 be A.

Point A is located 5 units to the right of $0$ on the number line.

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(b) – 10

Let the point representing -10 be B.

Point B is located 10 units to the left of $0$ on the number line.

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(c) + 8

Let the point representing +8 be C.

Point C is located 8 units to the right of $0$ on the number line.

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(d) – 1

Let the point representing -1 be D.

Point D is located 1 unit to the left of $0$ on the number line.

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(e) – 6

Let the point representing -6 be E.

Point E is located 6 units to the left of $0$ on the number line.

First, let's observe the vertical number line and determine the integer value for each marked point. We are given that point O represents 0 and point D represents +8. Since there are 8 units (marks) between O and D, each mark represents one unit.

By counting the units from O (zero):

Points above O (Positive Integers):

• Point A is 2 units above O, so A represents +2.
• Point B is 4 units above O, so B represents +4.
• Point C is 5 units above O, so C represents +5.
• Point D is 8 units above O, so D represents +8.

Points below O (Negative Integers):

• Point H is 2 units below O, so H represents -2.
• Point G is 6 units below O, so G represents -6.
• Point F is 8 units below O, so F represents -8.
• Point E is 10 units below O, so E represents -10.

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(a) If point D is + 8, then which point is – 8?

If point D represents +8, then the point representing -8 will be at the same distance (8 units) from 0 but in the opposite (downward) direction.

Based on our analysis of the number line, the point that is 8 units below 0 is point F.

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(b) Is point G a negative integer or a positive integer?

Point G is located below the zero mark (O). All points below zero on a vertical number line represent negative integers.

Therefore, point G is a negative integer.

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(c) Write integers for points B and E.

From our analysis of the number line:

Point B represents the integer +4.

Point E represents the integer -10.

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(d) Which point marked on this number line has the least value?

The least value on a vertical number line is the point that is the lowest. The lowest marked point on this number line is E.

The value of point E is -10, which is the smallest integer among all the marked points.

Therefore, point E has the least value.

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(e) Arrange all the points in decreasing order of value.

Decreasing order means arranging from the largest value to the smallest value. On a vertical number line, this corresponds to arranging the points from top to bottom.

The order of the points from top to bottom is:

D, C, B, A, O, H, G, F, E

(a) Write the temperatures of these places in the form of integers in the blank column.

Temperatures "below $0^\circ C$" are represented by negative integers, and temperatures "above $0^\circ C$" are represented by positive integers.

Place	Temperature Description	Temperature as Integer ($^\circ C$)
Siachin	10°C below 0°C	$-10$
Shimla	2°C below 0°C	$-2$
Ahmedabad	30°C above 0°C	$+30$
Delhi	20°C above 0°C	$+20$
Srinagar	5°C below 0°C	$-5$

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(b) Following is the number line representing the temperature in degree Celsius. Plot the name of the city against its temperature.

We plot each city on the number line according to its integer temperature:

• Siachin: $-10^\circ C$. Locate the mark labeled $-10$ on the number line and write "Siachin" there.
• Shimla: $-2^\circ C$. Locate the position corresponding to $-2$ (which is 2 units to the right of $-5$) and write "Shimla" there.
• Ahmedabad: $+30^\circ C$. Locate the mark labeled $30$ on the number line and write "Ahmedabad" there.
• Delhi: $+20^\circ C$. Locate the mark labeled $20$ on the number line and write "Delhi" there.
• Srinagar: $-5^\circ C$. Locate the mark labeled $-5$ on the number line and write "Srinagar" there.

The cities are plotted on the number line according to their respective temperatures.

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(c) Which is the coolest place?

The coolest place has the lowest temperature. We need to find the minimum value among the temperatures: $-10^\circ C$, $-2^\circ C$, $+30^\circ C$, $+20^\circ C$, $-5^\circ C$.

Comparing the integers: $-10, -2, 30, 20, -5$.

The smallest integer is $-10$.

The place with the temperature $-10^\circ C$ is Siachin.

Therefore, Siachin is the coolest place.

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(d) Write the names of the places where temperatures are above 10°C.

We need to find the places whose temperature $T$ satisfies $T > +10^\circ C$.

Let's examine the temperatures:

• Siachin: $-10^\circ C$ (Not above $+10^\circ C$)
• Shimla: $-2^\circ C$ (Not above $+10^\circ C$)
• Ahmedabad: $+30^\circ C$ (Above $+10^\circ C$, since $30 > 10$)
• Delhi: $+20^\circ C$ (Above $+10^\circ C$, since $20 > 10$)
• Srinagar: $-5^\circ C$ (Not above $+10^\circ C$)

The places with temperatures above $+10^\circ C$ are Ahmedabad and Delhi.

Concept:

On a standard horizontal number line, the values of integers increase as we move from left to right. Therefore, for any pair of distinct integers, the larger integer is always located to the right of the smaller integer.

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(a) 2, 9

We compare the integers $2$ and $9$.

Since $9 > 2$, the number 9 is to the right of $2$ on the number line.

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(b) – 3, – 8

We compare the integers $-3$ and $-8$.

Since $-3 > -8$, the number -3 is to the right of $-8$ on the number line.

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(c) 0, – 1

We compare the integers $0$ and $-1$.

Since $0 > -1$, the number 0 is to the right of $-1$ on the number line.

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(d) – 11, 10

We compare the integers $-11$ and $10$.

Since $10 > -11$ (any positive integer is greater than any negative integer), the number 10 is to the right of $-11$ on the number line.

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(e) – 6, 6

We compare the integers $-6$ and $6$.

Since $6 > -6$, the number 6 is to the right of $-6$ on the number line.

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(f) 1, – 100

We compare the integers $1$ and $-100$.

Since $1 > -100$, the number 1 is to the right of $-100$ on the number line.

Concept:

We need to find all integers $x$ that lie strictly between the two given integers. The integers should be listed in increasing order (from smallest to largest).

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(a) 0 and – 7

We are looking for integers $x$ such that $-7 < x < 0$.

Starting from the integer just greater than $-7$ and moving towards $0$, the integers are $-6, -5, -4, -3, -2, -1$.

These are already in increasing order.

The integers are: $-6, -5, -4, -3, -2, -1$

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(b) – 4 and 4

We are looking for integers $x$ such that $-4 < x < 4$.

Starting from the integer just greater than $-4$ and moving towards $4$, the integers are $-3, -2, -1, 0, 1, 2, 3$.

These are already in increasing order.

The integers are: $-3, -2, -1, 0, 1, 2, 3$

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(c) – 8 and – 15

We are looking for integers $x$ such that $-15 < x < -8$.

Starting from the integer just greater than $-15$ and moving towards $-8$, the integers are $-14, -13, -12, -11, -10, -9$.

These are already in increasing order.

The integers are: $-14, -13, -12, -11, -10, -9$

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(d) – 30 and – 23

We are looking for integers $x$ such that $-30 < x < -23$.

Starting from the integer just greater than $-30$ and moving towards $-23$, the integers are $-29, -28, -27, -26, -25, -24$.

These are already in increasing order.

The integers are: $-29, -28, -27, -26, -25, -24$

(a) Write four negative integers greater than – 20.

Integers greater than $-20$ are located to the right of $-20$ on the number line.

We need to find negative integers $x$ such that $x > -20$.

Examples of integers greater than $-20$ are $-19, -18, -17, -16, -15, \dots, -1$.

We need to list any four of these negative integers.

Four such negative integers are: $-19, -18, -17, -16$.

(Other possible answers exist, e.g., $-5, -10, -1, -12$)

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(b) Write four integers less than – 10.

Integers less than $-10$ are located to the left of $-10$ on the number line.

We need to find integers $x$ such that $x < -10$.

Examples of integers less than $-10$ are $-11, -12, -13, -14, -15, \dots$.

We need to list any four of these integers.

Four such integers are: $-11, -12, -13, -14$.

(Other possible answers exist, e.g., $-20, -50, -100, -11$)

(a) – 8 is to the right of – 10 on a number line.

On a number line, larger numbers are located to the right of smaller numbers.

Comparing $-8$ and $-10$, we find that $-8 > -10$.

Therefore, $-8$ is indeed to the right of $-10$.

The statement is True (T).

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(b) – 100 is to the right of – 50 on a number line.

Comparing $-100$ and $-50$, we find that $-100 < -50$.

Since $-100$ is smaller than $-50$, it is located to the left of $-50$ on the number line.

The statement is False (F).

Corrected statement: – 100 is to the left of – 50 on a number line. (Alternatively: -50 is to the right of -100).

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(c) Smallest negative integer is – 1.

Negative integers are integers less than 0: $\dots, -4, -3, -2, -1$.

On the number line, integers decrease as we move to the left. The negative integers extend infinitely to the left.

There is no smallest negative integer. $-1$ is the largest negative integer.

The statement is False (F).

Corrected statement: The largest negative integer is – 1.

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(d) – 26 is greater than – 25.

Comparing $-26$ and $-25$, we find that $-26 < -25$.

Therefore, $-26$ is not greater than $-25$; it is less than $-25$.

The statement is False (F).

Corrected statement: – 26 is less than – 25. (Alternatively: -25 is greater than -26).

(a) Which number will we reach if we move 4 numbers to the right of – 2.

Starting at $-2$ and moving 4 numbers to the right means we are adding 4.

Final position = $-2 + 4 = 2$.

We will reach the number 2.

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(b) Which number will we reach if we move 5 numbers to the left of 1.

Starting at $1$ and moving 5 numbers to the left means we are subtracting 5.

Final position = $1 - 5 = -4$.

We will reach the number -4.

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(c) If we are at – 8 on the number line, in which direction should we move to reach – 13?

We start at $-8$ and want to reach $-13$. Since $-13$ is a smaller number than $-8$, it is located to the left of $-8$ on the number line.

Therefore, we should move in the left direction.

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(d) If we are at – 6 on the number line, in which direction should we move to reach – 1?

We start at $-6$ and want to reach $-1$. Since $-1$ is a larger number than $-6$, it is located to the right of $-6$ on the number line.

Therefore, we should move in the right direction.

Solution (a): 4 more than –1

To find "4 more than -1", we start at -1 on the number line and move 4 steps to the right (the direction for 'more').

As shown on the number line, starting at -1 and moving 4 steps to the right brings us to 3.

Mathematically, this is calculated as:

$-1 + 4 = 3$

So, 4 more than $-1$ is 3.

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Solution (b): 5 less than 3

To find "5 less than 3", we start at 3 on the number line and move 5 steps to the left (the direction for 'less').

As shown on the number line, starting at 3 and moving 5 steps to the left brings us to -2.

Mathematically, this is calculated as:

$3 - 5 = -2$

So, 5 less than $3$ is -2.

Solution:

We need to find the sum of the integers: $(– 9) + (+ 4) + (– 6) + (+ 3)$.

First, we can rewrite the expression by simplifying the signs:

$-9 + 4 - 6 + 3$

To make the calculation easier, we can group the positive integers and the negative integers together.

Positive integers: $+4$ and $+3$.

Negative integers: $-9$ and $-6$.

Rearranging the terms:

$(+4) + (+3) + (-9) + (-6)$

or

$4 + 3 - 9 - 6$

Now, sum the positive integers:

$4 + 3 = 7$

Next, sum the negative integers:

$-9 - 6 = -(9 + 6) = -15$

Finally, combine the sum of positives and the sum of negatives:

$7 + (-15)$

This is equivalent to:

$7 - 15$

Performing the subtraction:

$7 - 15 = -8$

Therefore, the sum is $\mathbf{-8}$.

Solution:

We need to find the value of the expression: $(30) + (– 23) + (– 63) + (+ 55)$.

First, simplify the signs:

$30 - 23 - 63 + 55$

Next, group the positive integers and the negative integers:

Positive integers: $30$ and $55$.

Negative integers: $-23$ and $-63$.

Rearrange the terms:

$(30 + 55) + (-23 - 63)$

Calculate the sum of the positive integers:

$30 + 55 = 85$

Calculate the sum of the negative integers:

$-23 - 63 = -(23 + 63) = -86$

Now, combine the results:

$85 + (-86)$

This is equivalent to:

$85 - 86$

Performing the subtraction:

$85 - 86 = -1$

Therefore, the value of the expression is $\mathbf{-1}$.

Solution:

We need to find the sum of the integers: $(-10)$, $(92)$, $(84)$, and $(-15)$.

The expression for the sum is:

$(-10) + (92) + (84) + (-15)$

Simplify the signs:

$-10 + 92 + 84 - 15$

Group the positive integers and the negative integers:

Positive integers: $92$ and $84$.

Negative integers: $-10$ and $-15$.

Rearrange the terms:

$(92 + 84) + (-10 - 15)$

Calculate the sum of the positive integers:

$92 + 84 = 176$

Calculate the sum of the negative integers:

$-10 - 15 = -(10 + 15) = -25$

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Now, combine the results:

$176 + (-25)$

This is equivalent to:

$176 - 25$

Performing the subtraction:

$176 - 25 = 151$

Therefore, the sum is $\mathbf{151}$.

(a) 3 more than 5

Start at 5 on the number line and move 3 steps to the right.

The integer is 8.

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(b) 5 more than –5

Start at -5 on the number line and move 5 steps to the right.

The integer is 0.

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(c) 6 less than 2

Start at 2 on the number line and move 6 steps to the left.

The integer is -4.

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(d) 3 less than –2

Start at -2 on the number line and move 3 steps to the left.

The integer is -5.

(a) 9 + (– 6)

Start at 9 on the number line. Adding -6 means moving 6 steps to the left.

We reach the integer 3. So, $9 + (-6) = 3$.

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(b) 5 + (– 11)

Start at 5 on the number line. Adding -11 means moving 11 steps to the left.

We reach the integer -6. So, $5 + (-11) = -6$.

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(c) (– 1) + (– 7)

Start at -1 on the number line. Adding -7 means moving 7 steps to the left.

We reach the integer -8. So, $(-1) + (-7) = -8$.

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(d) (– 5) + 10

Start at -5 on the number line. Adding 10 means moving 10 steps to the right.

We reach the integer 5. So, $(-5) + 10 = 5$.

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(e) (– 1) + (– 2) + (– 3)

Start at -1. First, move 2 steps to the left to add -2, reaching -3. Then, from -3, move 3 steps to the left to add -3.

We reach the integer -6. So, $(-1) + (-2) + (-3) = -6$.

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(f) (– 2) + 8 + (– 4)

Start at -2. First, move 8 steps to the right to add 8, reaching 6. Then, from 6, move 4 steps to the left to add -4.

We reach the integer 2. So, $(-2) + 8 + (-4) = 2$.

(a) 11 + (– 7)

Adding a negative integer is the same as subtracting its positive counterpart.

$11 + (-7) = 11 - 7$

$11 - 7 = 4$

So, $11 + (-7) = \mathbf{4}$.

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(b) (– 13) + (+ 18)

This is the same as $-13 + 18$.

Since the integers have different signs, we subtract their absolute values: $|18| - |-13| = 18 - 13 = 5$.

The sign of the result is the same as the sign of the integer with the larger absolute value, which is $+18$.

So, $(-13) + (+18) = \mathbf{5}$.

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(c) (– 10) + (+ 19)

This is the same as $-10 + 19$.

Since the integers have different signs, we subtract their absolute values: $|19| - |-10| = 19 - 10 = 9$.

The sign of the result is the same as the sign of the integer with the larger absolute value, which is $+19$.

So, $(-10) + (+19) = \mathbf{9}$.

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(d) (– 250) + (+ 150)

This is the same as $-250 + 150$.

Since the integers have different signs, we subtract their absolute values: $|-250| - |+150| = 250 - 150 = 100$.

The sign of the result is the same as the sign of the integer with the larger absolute value, which is $-250$.

So, $(-250) + (+150) = \mathbf{-100}$.

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(e) (– 380) + (– 270)

This is the same as $-380 - 270$.

Since both integers are negative, we add their absolute values: $|-380| + |-270| = 380 + 270 = 650$.

The sign of the result is negative.

So, $(-380) + (-270) = \mathbf{-650}$.

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(f) (– 217) + (– 100)

This is the same as $-217 - 100$.

Since both integers are negative, we add their absolute values: $|-217| + |-100| = 217 + 100 = 317$.

The sign of the result is negative.

So, $(-217) + (-100) = \mathbf{-317}$.

(a) 137 and – 354

We need to calculate the sum: $137 + (-354)$.

This is equivalent to $137 - 354$.

Since the numbers have opposite signs, we subtract their absolute values: $|-354| - |137| = 354 - 137$.

$354 - 137 = 217$.

The result takes the sign of the number with the larger absolute value, which is $-354$.

So, the sum is $\mathbf{-217}$.

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(b) – 52 and 52

We need to calculate the sum: $(-52) + 52$.

An integer and its opposite (additive inverse) always sum to zero.

$-52 + 52 = 0$.

So, the sum is $\mathbf{0}$.

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(c) – 312, 39 and 192

We need to calculate the sum: $(-312) + 39 + 192$.

First, let's add the positive integers: $39 + 192 = 231$.

Now the expression becomes: $(-312) + 231$.

This is equivalent to $-312 + 231$.

Since the numbers have opposite signs, we subtract their absolute values: $|-312| - |231| = 312 - 231$.

$312 - 231 = 81$.

The result takes the sign of the number with the larger absolute value, which is $-312$.

So, the sum is $\mathbf{-81}$.

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(d) – 50, – 200 and 300

We need to calculate the sum: $(-50) + (-200) + 300$.

First, let's add the negative integers: $(-50) + (-200) = -50 - 200 = -250$.

Now the expression becomes: $-250 + 300$.

Since the numbers have opposite signs, we subtract their absolute values: $|300| - |-250| = 300 - 250$.

$300 - 250 = 50$.

The result takes the sign of the number with the larger absolute value, which is $300$.

So, the sum is $\mathbf{50}$.

(a) (– 7) + (– 9) + 4 + 16

First, simplify the expression:

$-7 - 9 + 4 + 16$

Group the positive and negative integers:

Positive integers: $4, 16$. Sum = $4 + 16 = 20$.

Negative integers: $-7, -9$. Sum = $-7 - 9 = -(7 + 9) = -16$.

Combine the sums:

$20 + (-16) = 20 - 16$

$20 - 16 = 4$

Therefore, the sum is $\mathbf{4}$.

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(b) (37) + (– 2) + (– 65) + (– 8)

First, simplify the expression:

$37 - 2 - 65 - 8$

Group the positive and negative integers:

Positive integer: $37$.

Negative integers: $-2, -65, -8$. Sum = $-2 - 65 - 8 = -(2 + 65 + 8) = -(67 + 8) = -75$.

Combine the sums:

$37 + (-75) = 37 - 75$

To calculate $37 - 75$, we subtract the smaller absolute value from the larger absolute value: $75 - 37 = 38$.

The result takes the sign of the number with the larger absolute value ($-75$), which is negative.

$37 - 75 = -38$

Therefore, the sum is $\mathbf{-38}$.

Solution:

We need to calculate the value of $-8 - (-10)$.

First, we know that subtracting a negative integer is the same as adding its positive counterpart (the additive inverse).

So, $-8 - (-10) = -8 + 10$.

Now, we can use the number line to find the value of $-8 + 10$.

1. Start at the first number, which is $\mathbf{-8}$, on the number line.

2. The operation is addition ($+$), and the number being added is positive ($+10$). Adding a positive number means moving to the right on the number line.

3. We need to move $\mathbf{10}$ steps to the right from $-8$.

Let's trace the movement:

Starting at $-8$:

1 step right: $-7$

2 steps right: $-6$

3 steps right: $-5$

4 steps right: $-4$

5 steps right: $-3$

6 steps right: $-2$

7 steps right: $-1$

8 steps right: $0$

9 steps right: $1$

10 steps right: $2$

After moving 10 steps to the right from $-8$, we reach the number $\mathbf{2}$.

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Therefore, $-8 - (-10) = -8 + 10 = \mathbf{2}$.

Solution:

We need to subtract the integer $(-4)$ from the integer $(-10)$.

This operation can be written as:

$(-10) - (-4)$

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Subtracting a negative integer is the same as adding the additive inverse (the positive version) of that integer.

The additive inverse of $-4$ is $+4$.

So, the expression becomes:

$(-10) + 4$

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Now we need to calculate the sum $-10 + 4$.

Since the integers have different signs, we find the difference between their absolute values: $|-10| - |4| = 10 - 4 = 6$.

The result takes the sign of the integer with the larger absolute value, which is $-10$. Therefore, the result is negative.

$-10 + 4 = -6$.

Thus, subtracting $(-4)$ from $(-10)$ gives $\mathbf{-6}$.

Solution:

We need to subtract the integer $(+3)$ from the integer $(-3)$.

The expression for this operation is:

$(-3) - (+3)$

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Subtracting a positive integer is the same as adding its negative counterpart (additive inverse).

So, the expression becomes:

$(-3) + (-3)$

Or simply:

$-3 - 3$

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Now we need to add two negative integers, $-3$ and $-3$.

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

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

Sum of absolute values: $3 + 3 = 6$.

Keep the negative sign.

$-3 - 3 = -6$.

Thus, subtracting $(+3)$ from $(-3)$ gives $\mathbf{-6}$.

(a) 35 – (20)

This is a straightforward subtraction of positive integers.

$35 - 20 = 15$

Result: 15

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(b) 72 – (90)

This is equivalent to $72 - 90$.

We can rewrite this as $72 + (-90)$.

Subtract the absolute values: $|-90| - |72| = 90 - 72 = 18$.

The sign of the result is the same as the number with the larger absolute value ($-90$), so the result is negative.

$72 - 90 = -18$

Result: -18

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(c) (– 15) – (– 18)

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

$(-15) - (-18) = -15 + 18$

Subtract the absolute values: $|18| - |-15| = 18 - 15 = 3$.

The sign of the result is the same as the number with the larger absolute value ($+18$), so the result is positive.

$-15 + 18 = 3$

Result: 3

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(d) (–20) – (13)

This is equivalent to $-20 - 13$.

We are adding two negative values (or subtracting a positive from a negative).

Add the absolute values: $|-20| + |13| = 20 + 13 = 33$.

The result is negative.

$-20 - 13 = -33$

Result: -33

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(e) 23 – (– 12)

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

$23 - (-12) = 23 + 12$

$23 + 12 = 35$

Result: 35

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(f) (–32) – (– 40)

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

$(-32) - (-40) = -32 + 40$

Subtract the absolute values: $|40| - |-32| = 40 - 32 = 8$.

The sign of the result is the same as the number with the larger absolute value ($+40$), so the result is positive.

$-32 + 40 = 8$

Result: 8

(a) (– 3) + (– 6) ______ (– 3) – (– 6)

Evaluate the Left Hand Side (LHS):

$(-3) + (-6) = -3 - 6 = -9$

Evaluate the Right Hand Side (RHS):

$(-3) - (-6) = -3 + 6 = 3$

Compare the results: $-9$ and $3$.

Since $-9$ is less than $3$, we have $-9 < 3$.

Therefore, (– 3) + (– 6) < (– 3) – (– 6)

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(b) (– 21) – (– 10) _____ (– 31) + (– 11)

Evaluate the Left Hand Side (LHS):

$(-21) - (-10) = -21 + 10 = -11$

Evaluate the Right Hand Side (RHS):

$(-31) + (-11) = -31 - 11 = -42$

Compare the results: $-11$ and $-42$.

Since $-11$ is greater than $-42$, we have $-11 > -42$.

Therefore, (– 21) – (– 10) > (– 31) + (– 11)

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(c) 45 – (– 11) ______ 57 + (– 4)

Evaluate the Left Hand Side (LHS):

$45 - (-11) = 45 + 11 = 56$

Evaluate the Right Hand Side (RHS):

$57 + (-4) = 57 - 4 = 53$

Compare the results: $56$ and $53$.

Since $56$ is greater than $53$, we have $56 > 53$.

Therefore, 45 – (– 11) > 57 + (– 4)

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(d) (– 25) – (– 42) _____ (– 42) – (– 25)

Evaluate the Left Hand Side (LHS):

$(-25) - (-42) = -25 + 42 = 17$

Evaluate the Right Hand Side (RHS):

$(-42) - (-25) = -42 + 25 = -17$

Compare the results: $17$ and $-17$.

Since $17$ is greater than $-17$, we have $17 > -17$.

Therefore, (– 25) – (– 42) > (– 42) – (– 25)

(a) (– 8) + _____ = 0

The additive inverse of an integer is the number that, when added to the integer, results in zero. The additive inverse of $-8$ is $8$.

$(-8) + \mathbf{8} = 0$

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(b) 13 + _____ = 0

The additive inverse of $13$ is $-13$.

$13 + \mathbf{(-13)} = 0$

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(c) 12 + (– 12) = ____

This is the sum of an integer ($12$) and its additive inverse ($-12$). The sum is always zero.

$12 + (-12) = \mathbf{0}$

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(d) (– 4) + ____ = – 12

Let the unknown number be $x$. We have the equation $-4 + x = -12$.

To find $x$, we can add $4$ to both sides of the equation:

$x = -12 + 4$

$x = -8$

So, $(-4) + \mathbf{(-8)} = -12$.

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(e) ____ – 15 = – 10

Let the unknown number be $y$. We have the equation $y - 15 = -10$.

To find $y$, we can add $15$ to both sides of the equation:

$y = -10 + 15$

$y = 5$

So, $\mathbf{5} - 15 = -10$.

(a) (– 7) – 8 – (– 25)

First, simplify the expression. Subtracting a negative integer is the same as adding its positive counterpart.

$(– 7) – 8 – (– 25) = -7 - 8 + 25$

Combine the negative terms:

$-7 - 8 = -15$

Now the expression is:

$-15 + 25$

Calculate the sum:

$-15 + 25 = 10$

So, $(– 7) – 8 – (– 25) = \mathbf{10}$.

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(b) (– 13) + 32 – 8 – 1

The expression is:

$-13 + 32 - 8 - 1$

Group the positive and negative terms:

Positive term: $32$.

Negative terms: $-13, -8, -1$.

Sum the negative terms: $-13 - 8 - 1 = -(13 + 8 + 1) = -(21 + 1) = -22$.

Combine the sums:

$32 + (-22) = 32 - 22$

$32 - 22 = 10$

So, $(– 13) + 32 – 8 – 1 = \mathbf{10}$.

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(c) (– 7) + (– 8) + (– 90)

The expression is:

$-7 - 8 - 90$

Since all terms are negative, add their absolute values and keep the negative sign.

$-(7 + 8 + 90) = -(15 + 90) = -105$

So, $(– 7) + (– 8) + (– 90) = \mathbf{-105}$.

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(d) 50 – (– 40) – (– 2)

First, simplify the expression. Subtracting a negative integer is the same as adding its positive counterpart.

$50 – (– 40) – (– 2) = 50 + 40 + 2$

Add the terms:

$50 + 40 = 90$

$90 + 2 = 92$

So, $50 – (– 40) – (– 2) = \mathbf{92}$.