1 Lesson 2.2.1 Absolute Value. 2 Lesson 2.2.1 Absolute Value California Standard: Number Sense 2.5 Understand the meaning of the absolute value of a number;

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Lesson 2.2.1Lesson 2.2.1

Absolute ValueAbsolute Value

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Lesson

2.2.1Absolute ValueAbsolute Value

California Standard:Number Sense 2.5Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.

What it means for you:You’ll learn how to find the absolute value of a number, and use it in calculations.Key words:• absolute value• distance• opposite

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Absolute ValueAbsolute ValueLesson

2.2.1

You can think of the number “–5” as having two parts — a negative sign that tells you it’s less than zero, and “5,” which tells you its size, or how far from zero it is.

The absolute value of a number is just its size — it’s not affected by whether it’s greater or less than zero.

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Absolute Value is Distance From Zero

Lesson

2.2.1

The absolute value of a number is never negative — that’s because the absolute value describes how far the number is from zero on the number line.

The absolute value of a number is its distance from 0 on the number line.

It doesn’t matter if the number is to the left or to the right of zero — the distance can’t be negative.

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Opposites Have the Same Absolute Value

Lesson

2.2.1

Opposites are numbers that are the same distance from 0, but going in opposite directions.

Opposites have the same absolute value.

–5 and 5 are opposites:

So they each have an absolute value of 5.

0 1 2 3–1–2–3–4–5–6 4 5 6

Distance of 5 Distance of 5

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Absolute ValueAbsolute ValueLesson

2.2.1

A set of bars, | |, are used to represent absolute value.

For example, the expression:

|–10| means “the absolute value of negative ten.”

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Example 1

Solution follows…

Lesson

2.2.1

What is |3.25|? What is |–3.25|?

Solution

3.25 and –3.25 are opposites. They’re the same distance from 0, so they have the same absolute value.

So, |3.25| = |–3.25| = 3.25

0 1 2 3–1–2–3–4–5–6 4 5 6

Distance of 3.25 Distance of 3.25

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Find the values of the expressions in Exercises 1–8.

1. |12| 2. |–9|

3. |16| 4. |–1|

5. |1.7| 6. |–3.2|

7. |– | 8. |0|

In Exercises 9–12, say which is bigger.

9. |17| or |16| 10. |–2| or |–5|

11. |–9| or |8| 12. |–1| or |1|

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Guided Practice

Solution follows…

Lesson

2.2.1

12 9

16 1

1.7 3.2

01

2

|17|

|–9|

|–5|

They are the same size.

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Absolute Value Equations Often Have Two Solutions

Lesson

2.2.1

Think about the equation |x| = 2.

The absolute value of x is 2, so you know that x is 2 units away from 0 on the number line, but you don’t know in which direction. x could be 2, but it could also be –2.

You can show the two possibilities like this:

0 1 2–1–2

2 units 2 units

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Example 2

Solution follows…

Lesson

2.2.1

Solve |z| = 3.

Solution

z can be either 3 or –3.

0 1 2 3–1–2–3–4–5–6 4 5 6

3 units 3 units

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Guided Practice

Solution follows…

Lesson

2.2.1

Give the solutions to the equations in Exercises 13–16.

13. |a| = 1 14. |r| = 4

15. |q| = 6 16. |g| = 7

a = 1 or –1

q = 6 or –6 g = 7 or –7

r = 4 or –4

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Treat Absolute Value Signs Like Parentheses

Lesson

2.2.1

You should treat absolute value bars like parentheses when you’re deciding what order to do the operations in.

Work out what’s inside them first, then take the absolute value of that.

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Example 3

Solution follows…

Lesson

2.2.1

What is the value of |7 – 3| + |4 – 6|?

Solution

|7 – 3| + |4 – 6|

= |4| + |–2|

= 4 + 2

= 6

Simplify whatever is inside the absolute value signs

Find the absolute values

Simplify the expression

Write out the expression

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Guided Practice

Solution follows…

Lesson

2.2.1

Evaluate the expressions in Exercises 17–22.

17. |1 – 3| – |2 + 2| 18. |2 – 7| + |0 – 6|

19. –|5 – 6| 20. |–8| × |2 – 3|

21. 2 × |4 – 6| 22. |7 – 2| ÷ |1 – 6|

–2 11

–1 8

4 1

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Independent Practice

Solution follows…

Lesson

2.2.1

Evaluate the expressions in Exercises 1–4.1. |–45| 2. |6|

3. |–0.6| 4. | |

5. Let x and y be two integers. The absolute value of y is larger than the absolute value of x. Which of the two integers is further from 0?

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65

6

45

0.6

6

y

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Solution follows…

Lesson

2.2.1

Show the solutions of the equations in Exercises 6–9 on number lines.

6. |u| = 3

7. |d| = 9

8. |x| = 5

9. |w| = 15

0 1 2 3–1–2–3–4–5 4 5

0 1 2 3–1–2–3–4 4

0 3 6 9–3–6–9–12 12

0 3 6 9–3–6–9–12–15 12 15

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Solution follows…

Lesson

2.2.1

Show the solutions of the equations in Exercises 10–13 on number lines.

10. |y| = 4

11. |v| = 1

12. |k| = 16

13. |z| = 7

0 1 2 3–1–2–3–4–5 4 5

0 1 2 3–1–2–3

0 4 8 12–4–8–12–16–20 16 20

0 2 4 6–2–4–6

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Solution follows…

Lesson

2.2.1

In Exercises 14–19, say which is bigger.14. |–6| or |–1| 15. |3| or |–5|

16. |2 – 2| or |5 – 8| 17. |6 – 8| or |2 – 1|

18. |3 – 2| or |–5| 19. |11 + 1| or |–2 – 8|

Evaluate the expressions in Exercises 20–25.

20. |3 – 5| + |2 – 5| 21. |0 + 5| + |0 – 5|

22. |5 – 10| – |0 – 2| 23. |–1| × |3 – 3|

24. 8 × |1 – 4| 25. |2 – 8| ÷ |4 – 1|

|–6| |–5|

|5 – 8| |6 – 8|

|–5| |11 + 1|

5 10

3 0

24 2

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Solution follows…

Lesson

2.2.1

26. What is the sum of two different numbers that have the same absolute value? Explain your answer.

27. Is it always true that |y| < 2y when y is an integer? No. If y is negative or zero it isn’t true.

0. For two different numbers to have the same absolute value they must be “opposites,” e.g. 3 and –3.

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Round UpRound Up

The absolute value of a number is its distance from zero on a number line.

Lesson

2.2.1

If you see absolute value bars in an expression, work out what’s between them first — just like parentheses.

Absolute values are always positive. So if a number has a negative sign, get rid of it; if it doesn’t, then leave it alone.

1 Lesson 2.2.1 Absolute Value. 2 Lesson 2.2.1 Absolute Value California Standard: Number Sense 2.5 Understand the meaning of the absolute value of a number;

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