Integers and Absolute Value

Lesson 3-1

Identify and Graph Integers

Integers can be graphed on a number line. To graph an integer on the number line, draw

a dot on the line at its location.

Example 1

Write an integer for each situation.a. An average temperature of 5 degrees below normal

Because it represents below normal, the integer is -5

b. An average rainfall of 5 inches above normal.

Because it represents above normal, the integer is +5 or 5.

Got it? 1

Write an integer for each situation. a. 6 degrees above normal

+6

b. 2 inches below normal-2

Example 2

Graph the set of integers {4, -6, 0} on a number line.

Got it? 2

Graph each set of integers on a number line. a. {-2, 8, -7}

b. {-4, 10, -3, 7}

Absolute Value

On the number line in the box, notice that -5 and 5 are each 5 units from 0, even though they are on opposites sides of 0. Numbers that are the same distance from zero on the number line have the same absolute value.

Example 3

Evaluate each expression. a. -4

The graph of -4 is 4 units from 0.So, -4 = 4

b. -5 - 2-5 - 2

5 – 23

Got it? 3

Evaluate these expressions.a. 8

8

b. 2 + -35

c. -6 - 51

Example 4

Nick climbs 30 feet up a rock wall and then climbs 22 feet down to a landing area. The number of feet Nick climbs can be represented using the expression 30 + -22. How many feet does Nick climb?

30 + -22 = 30 + -22= 30 + 22

= 52

Add IntegersLesson 3-2

Key Concept:

Example 1

a. Find -3 + (-2).Start at 0. Move 3 units down to show -3.

From there, move 2 units down to show -2.

Example 1

b. Find -26 + (-17).-26 + (-17) = -43

Got it? 1

a. -5 + (-7)-12

b. -10 + (-4)-14

c. -14 + (-16)-30

Key Concept:

When you add integers with different signs, start at zero. Move right for positive integers. Move left for negative integers. So, the sum of p + q is located a distance q+ p.

Example 2

a. Find 5 + (-3)

So, 5 + (-3) = 2

b. Find -3 + 2

So, -3 + 2 = -1

Got it? 2

a. Find 6 + (-7)

-1

b. Find -15 + 19

4

Example 3

a. Find 7 + (-7)7 + (-7) = 0

b. Find -8 + 3-8 + 3 = -5

c. Find 2 + (-15) + (-2)2 + (-2) + (-15)

0 + (-15)= -15

Got it? 3

a. 10 + (-12)-2

b. -13 + 185

c. (-14) + (-6) + 6-14

Example 4

A roller coaster starts at point A. It goes up 20 feet, down 32 feet, and then up 16 feet to point B. Write an addition sentence to find the height at point B in relation to point A. Then find the sum and explain its meaning.

20 +(-32) + 16 = 20 + 16 + (-32)= 36 + (-32)

= 4

Point B is 4 feet higher than point A.

Got it? 4

The temperature is -3. An hour later, it drops 6 and 2 hours later it rises 4. Write an addition expression to describe this situation. Then find the sum and explain its meaning.

-3 + (-6) + 4 = -5

The new temperature is -5F.

Subtract IntegersLesson 3-3

Subtract Integers

Words: To subtract an integer, add its additive inverse.

Symbols: p – q = p + (-q)

Examples: 4 – 9 = 4 + (-9) 7 – (-10) = 7 + 10

When you subtract 7, the result is the same as adding its additive inverse, -7.

Example 1

a. Find 8 – 13. 8 – 13 = 8 + (-13)

= -5

b. Find -10 – 7. -10 – 7 = -10 + (-7)

= -27

Got it? 1

a. 6 – 12 -6

b. -20 – 15 -35

c. -22 – 26 -48

Example 2

a. Find 1 – (-2). 1 – (-2) = 1 + 2

= 3

b. Find -10 – (-7).-10 – (-7) = -10 + 7

= -3

Got it? 2

a. 4 – (-12)16

b. -15 – (-5)-10

c. 18 – (-6)24

Example 3

a. Evaluate x – y if x = -6 and y = -5x – y = -6 – (-5)

= -6 + 5= -1

b. Evaluate m – n if m = -15 and n= 8m – n = -15 – 8

= -15 + (-8)= -23

Got it? 3

Evaluate each expression if a = 5, b = -8, and c = -9.

a. b - 10-18

b. a – b 13

c. c – a -14

Example 4

The temperatures on the Moon vary from -173C to 127C. Find the difference between the maximum and minimum temperatures.

Subtract the lower temperatures from the higher temperature.

127 – (-173) = 127 + 173= 300.

The difference between the two temperatures is 300C.

Got it? 4

Brenda had a balance of -$52 in her account. The bank charged her a fee of $10 for having a negative balance. What is her new balance?

-$62

Multiply IntegersLesson 3-4

Multiply Integers with Different Signs

Words: The product of two integers with different signs is negative.

Examples: 6(-4) = -24 -5(7) = -35

Remember that multiplication is the same as repeated addition. 4(-3) = (-3) + (-3) + (-3) + (-3) = -12

Example 1

a. Find 3(-5).

3(-5) = -15 Different signs, negative

b. Find -6(8).

-6(8) = -48 Different signs, negative

Got it? 1

a. 9(-2) = -18

b. -7(4) = -28

Multiply Integers with the Same Signs

Words: The product of two integers with same signs is positive. Examples: 2(6) = 12 -10(-6) = 60

The product of two positive integers is positive. You can use a pattern to find the sign of products of two negative integers. Start with (2)(-3) = -6 and (1)(-3) = -3.

Multiply Integers with the Same Signs

Each product is 3 more than the previous. This pattern can also be shown on a number line.

If you extend the pattern, the next two products are (-3)(-3) = 9 and (-4)(-3) = 12.

Example 2

a. Find -11(-9)-11(-9) = 99 Same signs, positive

b. Find (-4)2

(-4)(-4) = 16 Same signs, positive

c. Find -3(-4)(-2)-3(-4) = 12 Same signs, positive

12(-2) = -24 Different signs, negative

Got it? 2

a. -12(-4)48

b. (-5)2

25

c. -7(-5)(-3)-105

Example 3

A submersible is diving from the surface of the water at a rate of 90 feet per minute. What is the depth of the submersible after 7 minutes.

The submersible descends 90 feet per minute. After 7 minutes, the vessel will be at 7(-90) or -

630 feet.

The submersible will be 630 feet below sea level.

Got it? 3

Mr. Simon’s bank automatically deducts a $4 monthly maintenance fee from his savings account. Write a multiplication expression to represent the maintenance fees for one year. Then find the product and explain its meaning.

12(-4) = -48Mr. Simon will have $48 deducted from his

account at the end of the year.

Divide IntegersLesson 3-5

Divide Integers with Different Signs

Words: The quotient of two integers with different signs is negative.

Examples: 33 (-11) = 3 -64 8 = -8

Example 1

a. Find 80 (-10).

80 (-10) = -8 Different signs, negative

b. Find .

-55 11 = -5 Different signs, negative

Example 2

Use the table to find the constant rate of change in centimeters per hour.

The height of the candle decreases by 2 centimeters each hour.

So the constant rate of change is -2 centimeters per hour.

Got it? 1 & 2

a. 20 (-4) = -5

b. = -9

c. -45 9 = -5

Divide Integers with the Same Signs

Words: The quotient of two integers with the same signs is positive.

Examples: 15 5 = 3 -64 (-8) = 8

Example 3

a. Find -14 (-7).

-14 (-7) = 2 Same signs, positive

b. Find .

-27 -3 = 9 Same signs, positive

Example 4

Evaluate -16 x if x = -4.-16 x

-16 -4 = 4

Same signs, positive

Got it? 3 & 4

a. -24 (-4) = b. -9 (-3) =

c. d. Evaluate a b if a = -33 and b = -3.

7 3

4

11

Example 5

One year, the estimated Australian koala population was 1,000,000. After 10 years, there were about 100,000 koalas. Find the average change in the koala population per year. Then explain its meaning.

The koala population has changed by -90,000 per year.

Got it? 5

The average temperature in January for North Pole, Alaska, is -24C. Use the expression to find this temperature in degrees Fahrenheit. Round to the nearest degree. Then explain its meaning.

-11F

-24C is about -11F.