Lesson The Absolute Value of a Number

Chapter 1

1-8AVocabulary

BIG IDEA BIG IDEA

Vocabularyabsolute value

magnitude

BIG IDEA The absolute value of a number tells you how far the number is from 0.

The absolute value of a number is its distance from 0 on a number line. The symbol for absolute value is two vertical lines 1 . The distance of 2 from 0 is 2, so 2 = 2. The distance of –7 from 0 is 7, so –7 = 7. The points and their distances are shown on the number line below.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2

2

7

3 4 5 6 7 8 9 10

You can easily fi nd the absolute value of any number, 0, positive, or negative, as Example 1 shows.

Example 1In the table below, � ll in the missing values.

NumberDistance of the number from 0

Absolute value of the number Absolute value = distance

a. 0 0 0 0 = 0

b. 5 1 __ 2 5 1 __ 2 ? ? = 5 1 __ 2

c. – 4.8 4.8 – 4.8 ? = 4.8

d. –150 ? –150 –150 = ?

e. 1429 ? ? ? = ?

Solution

a. 0 is 0 units away from 0. The absolute value of 0 is 0 . The absolute value of a number equals its distance from 0, so 0 = 0.

b. 5 1 __ 2 is 5 1 __ 2 units away from 0. The absolute value of 5 1 __ 2 is ? ,

so ? = 5 1 __ 2 .

c. – 4.8 is 4.8 units away from 0. The absolute value of – 4.8 is – 4.8 , so ? = 4.8.

GUIDEDGUIDED

Vocabulary

1 Some Uses of Integers and Fractions

Lesson 1-8A

d. –150 is ? units away from 0. The absolute value of –150 is –150 , so –150 = ? .

e. 1429 is ? units away from 0. The absolute value of 1429 is ? , so ? = ? .

Notice that a number can be negative, but the absolute value of a number is never negative.

QY

Example 2On a number line, show the two numbers whose absolute value is 6.

Solution Because –6 and 6 are both 6 units from 0, those are the two numbers whose absolute value is 6.

The term magnitude is sometimes used instead of absolute value. For instance, an error may be zero, positive, or negative, but we may speak of the magnitude of the error, which is never negative.

Example 3A line segment is 6 inches long. Mia measured its length as 5 3 __ 4 inches. André measured its length as 6 1 __ 8 inches.

a. Find the error in measurement for each student. (Use a negative number to represent an error in measurement that produces a value below the actual length.)

b. Find the magnitude of each student’s error in measurement.

c. Whose measurement was more accurate?

Solution

a. Mia‛s error is 5 3 __ 4 in. - 6 in. = – 1 __ 4 in.

André‛s error is 6 1 __ 8 in. - 6 in. = 1 __ 8 in.

b. The magnitude of Mia‛s error is – 1 __ 4 = 1 __ 4 in.

The magnitude of André‛s error is 1 __ 8 = 1 __ 8 in.

c. 1 __ 8 < – 1 __ 4 . André‛s measurement was more accurate.

QY

Find each absolute value. a. 10 b. –10

c. – 3 1 __ 2

d. 2500

QY

Find each absolute value. a. 10

b. –10

c. – 3 1__2

d. 2500

QY

Find each absolute value. a. 10 b. –10

c. – 3 1 __ 2

d. 2500

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

6

6

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

6

6

The Absolute Value of a Number 2

Chapter 1

Notice that a smaller number may have a greater absolute value than a larger number. This can happen only when the smaller number is negative. For instance,

–30 is smaller than 20

but –30 = 30, which is greater than 20 = 20.

This is signifi cant in real-world contexts such as account balances. Suppose that owing $30 is represented by a balance of –30. If you have $20 in another account, then what you have is not enough to pay off your debt because 20 < –30 .

QuestionsCOVERING THE IDEAS

1. Multiple Choice Which of the following describes the absolute value of a number?

A It is always positive. B It is always negative. C It is always either positive or zero. D It may be positive, negative, or zero.

In 2–4, a number is given.a. What is the distance of the number from 0?b. What is the absolute value of the number?

2. –39 3. 2.9 4. – 4

5. Which two numbers have an absolute value of 12? Justify your answer.

6. Graph the two numbers whose distance from 0 is 3.25 on a horizontal number line.

7. What is another term for absolute value?

8. People were asked to estimate how many beans were in a bean bag. There were actually 324 beans in the bag. Will guessed 250 and May guessed 400.

a. What was the error in Will’s guess? b. What was the magnitude of the error in Will’s guess? c. What was the error in May’s guess? d. What was the magnitude of the error in May’s guess? e. Whose guess, Will’s or May’s, was off by more?

APPLYING THE MATHEMATICS

In 9–11, True or False.

9. The absolute value of a positive number is equal to that number.

10. The absolute value of a negative number is equal to that number.

11. The absolute value of zero is not a number.

3 Some Uses of Integers and Fractions

Lesson 1-8A

In 12–15, numbers s and t are given.a. Tell whether s > t.b. Tell whether s > t .

12. s = 50, t = –51 13. s = – 1 __ 2 , t = 1 __ 3

14. s = 4.09, t = 4.089 15. s = – 4 1 __ 2 , t = –5Fill in the Blank In 16–19, write >, <, or = to make the sentence true.

16. 3 ? –3

17. –5 ? –5

18. 3 __ 4 ? – 6 __ 8

19. –62 + 62 ? –70 + 70

In 20–23, evaluate the expression.

20. 3 + –9 + 0

21. 25.862 + –13.06 + –8.5

22. 2.56 - –1.34

23. 0 + 103 1 __ 8

24. a. Copy and complete the table below showing temperature change in a 24-hour period.

Time interval Description Number Magnitude

midnight to 4:00 A.M. temperature fell 5° –5 5

4:00 A.M. to 8:00 A.M. temperature did not change ? ?

8:00 A.M. to noon temperature rose 8° ? ?

noon to 4:00 P.M. temperature rose 7° ? ?

4:00 P.M. to 8:00 P.M. temperature fell 3° ? ?

8:00 P.M. to midnight temperature fell 10° ? ?

b. During which four-hour time period did the temperature change the most?

c. Was the temperature greater at the beginning or the end of the day?

25. In a large city, the rapid transit trains operate both above ground and below ground. On one line, the tracks are about 35 feet above ground on an elevated portion and are about 20 feet below ground during the subway portion. Use absolute value notation and an inequality symbol to compare the magnitudes of those values.

QY ANSWER

a. 10

b. 10

c. 3 1 __ 2

d. 2500

The Absolute Value of a Number 4