1. Absolute Value
Lesson 2.2.1

2. Absolute Value 2.2.1 California Standard: What it means for you:
Lesson
2.2.1
Absolute Value
California Standard:
Number Sense 2.5
Understand 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

3. Lesson
2.2.1
Absolute Value
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.

4. Absolute Value 2.2.1 Absolute Value is Distance From Zero
Lesson
2.2.1
Absolute Value
Absolute Value is Distance From Zero
The absolute value of a number is its distance from 0 on the number line.
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.
It doesn’t matter if the number is to the left or to the right of zero — the distance can’t be negative.

5. Absolute Value 2.2.1 Opposites Have the Same Absolute Value
Lesson
2.2.1
Absolute Value
Opposites Have the Same Absolute Value
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: 1 2 3 –1 –2 –3 –4 –5 –6 4 5 6
Distance of 5
So they each have an absolute value of 5.

6. Lesson
2.2.1
Absolute Value
A set of bars, | |, are used to represent absolute value.
For example, the expression:
|–10| means “the absolute value of negative ten.”

7. Absolute Value 2.2.1 What is |3.25|? What is |–3.25|? Solution
Lesson
2.2.1
Absolute Value
Example 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. 1 2 3 –1 –2 –3 –4 –5 –6 4 5 6
Distance of 3.25
So, |3.25| = |–3.25| = 3.25
Solution follows…

8. Absolute Value 2.2.1 Guided Practice
Lesson
2.2.1
Absolute Value
Guided Practice
Find the values of the expressions in Exercises 1–8.
1. |12| |–9|
3. |16| |–1|
5. |1.7| 6. |–3.2|
7. |– | 8. |0|
In Exercises 9–12, say which is bigger.
9. |17| or |16| |–2| or |–5|
11. |–9| or |8| |–1| or |1| 1 2 12 9 16 1
1.7
3.2 1 2
|17|
|–5|
|–9|
They are the same size.
Solution follows…

9. Absolute Value 2.2.1 Absolute Value Equations Often Have Two Solutions
Lesson
2.2.1
Absolute Value
Absolute Value Equations Often Have Two Solutions
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: 1 2 –1 –2
2 units

10. Absolute Value 2.2.1 Solve |z| = 3. Solution z can be either 3 or –3.
Lesson
2.2.1
Absolute Value
Example 2
Solve |z| = 3.
Solution
z can be either 3 or –3. 1 2 3 –1 –2 –3 –4 –5 –6 4 5 6
3 units
Solution follows…

11. Absolute Value 2.2.1 Guided Practice
Lesson
2.2.1
Absolute Value
Guided Practice
Give the solutions to the equations in Exercises 13–16.
13. |a| = |r| = 4
15. |q| = |g| = 7
a = 1 or –1
r = 4 or –4
q = 6 or –6
g = 7 or –7
Solution follows…

12. Absolute Value 2.2.1 Treat Absolute Value Signs Like Parentheses
Lesson
2.2.1
Absolute Value
Treat Absolute Value Signs Like Parentheses
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.

13. Absolute Value 2.2.1 What is the value of |7 – 3| + |4 – 6|? Solution
Lesson
2.2.1
Absolute Value
Example 3
What is the value of |7 – 3| + |4 – 6|?
Solution
|7 – 3| + |4 – 6|
Write out the expression
Simplify whatever is inside the absolute value signs
= |4| + |–2|
= 4 + 2
Find the absolute values
= 6
Simplify the expression
Solution follows…

14. Absolute Value 2.2.1 Guided Practice
Lesson
2.2.1
Absolute Value
Guided Practice
Evaluate the expressions in Exercises 17–22.
17. |1 – 3| – |2 + 2| |2 – 7| + |0 – 6|
19. –|5 – 6| |–8| × |2 – 3|
21. 2 × |4 – 6| |7 – 2| ÷ |1 – 6| –2 11 –1 8 4 1
Solution follows…

15. Absolute Value 2.2.1 Independent Practice
Lesson
2.2.1
Absolute Value
Independent Practice
Evaluate the expressions in Exercises 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? 45 6 5 6 5 6
0.6 y
Solution follows…

16. Absolute Value 2.2.1 Independent Practice
Lesson
2.2.1
Absolute Value
Independent Practice
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 1 2 3 –1 –2 –3 –4 4 3 6 9 –3 –6 –9
–12 12 1 2 3 –1 –2 –3 –4 –5 4 5 3 6 9 –3 –6 –9
–12
–15 12 15
Solution follows…

17. Absolute Value 2.2.1 Independent Practice
Lesson
2.2.1
Absolute Value
Independent Practice
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 1 2 3 –1 –2 –3 –4 –5 4 5 1 2 3 –1 –2 –3 4 8 12 –4 –8
–12
–16
–20 16 20 2 4 6 –2 –4 –6
Solution follows…

18. Absolute Value 2.2.1 Independent Practice
Lesson
2.2.1
Absolute Value
Independent Practice
In Exercises 14–19, say which is bigger |–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| |0 + 5| + |0 – 5|
22. |5 – 10| – |0 – 2| 23. |–1| × |3 – 3|
24. 8 × |1 – 4| |2 – 8| ÷ |4 – 1|
|–6|
|–5|
|5 – 8|
|6 – 8|
|–5|
|11 + 1| 5 10 3 24 2
Solution follows…

19. Absolute Value 2.2.1 Independent Practice
Lesson
2.2.1
Absolute Value
Independent Practice
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?
0. For two different numbers to have the same absolute value they must be “opposites,” e.g. 3 and –3.
No. If y is negative or zero it isn’t true.
Solution follows…

20. Absolute Value 2.2.1 Round Up
Lesson
2.2.1
Absolute Value
Round Up
The absolute value of a number is its distance from zero on a number line.
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.
If you see absolute value bars in an expression, work out what’s between them first — just like parentheses.