Absolute Value

Absolute value is used to represent the distance of a number from 0 on a number line. Since distance is always referred to as a nonnegative number, the absolute value of an expression is nonnegative.

On the number line below, both 4 and − 4 negative 4  are four units from zero. Therefore, | 4 | absolute value of 4 ,  and | − 4 | absolute value of negative 4 , end absolute value ,  are both equal to four.

When working with more complicated expressions, always remember to simplify within absolute value symbols first.

Example 1

Simplify each expression.

1. | 4 | + | − 19 | absolute value of 4 vertical line plus vertical line negative 19 , end absolute value ,

   | 4 | + | − 19 | = 4 + 19   = 23 table with 2 rows and 3 columns , row1 column 1 , absolute value of 4 vertical line plus vertical line negative 19 , end absolute value , , column 2 equals , column 3 4 plus 19 , row2 column 1 , , column 2 equals , column 3 23 , end table

2. | 4 − 8 | absolute value of 4 minus 8 , end absolute value ,

   | 4 − 8 | = | − 4 |   = 4 table with 2 rows and 3 columns , row1 column 1 , absolute value of 4 minus 8 , end absolute value , , column 2 equals , column 3 absolute value of negative 4 , end absolute value , , row2 column 1 , , column 2 equals , column 3 4 , end table

3. − 3 | − 7 − 4 | negative 3 vertical line negative 7 minus 4 vertical line

   − 3 | − 7 − 4 | = − 3 | − 11 |   = − 3 ⋅ 11   = − 33 table with 3 rows and 3 columns , row1 column 1 , negative 3 vertical line negative 7 minus 4 vertical line , column 2 equals , column 3 negative 3 vertical line negative 11 vertical line , row2 column 1 , , column 2 equals , column 3 negative 3 dot 11 , row3 column 1 , , column 2 equals , column 3 negative 33 , end table

To solve the absolute value equation | x | = a , vertical line x vertical line equals eh comma  find all the values x that are a units from 0 on a number line.

Example 2

Algebra Solve.

1. | x | = 7 x = 7  or  − 7 table with 2 rows and 3 columns , row1 column 1 , absolute value of bold italic x , , column 2 equals , column 3 7 , row2 column 1 , x , column 2 equals , column 3 7 , or , minus 7 , end table
2. | x | − 3 = 22 | x | − 3 = 22 | x | = 25 x = 25  or  − 25 table with 4 rows and 3 columns , row1 column 1 , vertical line bold italic x vertical line negative 3 , column 2 equals , column 3 22 , row2 column 1 , vertical line x vertical line negative 3 , column 2 equals , column 3 22 , row3 column 1 , absolute value of x , , column 2 equals , column 3 25 , row4 column 1 , x , column 2 equals , column 3 25 , or , minus 25 , end table

Exercises

Simplify each expression.

1. | − 8 | absolute value of negative 8 , end absolute value ,
2. | 11 | absolute value of 11 ,
3. | − 7 | + | 15 | absolute value of negative 7 vertical line plus vertical line 15 , end absolute value ,
4. | − 12 | − | − 12 | absolute value of negative 12 vertical line negative vertical line negative 12 , end absolute value ,
5. | − 5 | − | 10 | absolute value of negative 5 vertical line negative vertical line 10 , end absolute value ,
6. | − 4 | + | − 2 | absolute value of negative 4 vertical line plus vertical line negative 2 , end absolute value ,
7. 10 − | − 20 | 10 minus vertical line negative 20 vertical line
8. | − 9 | − 15 vertical line negative 9 vertical line negative 15
9. | 4 − 17 | absolute value of 4 minus 17 , end absolute value ,
10. | − 9 − 11 | absolute value of negative 9 minus 11 , end absolute value ,
11. 2 | − 21 + 16 | 2 vertical line negative 21 plus 16 vertical line
12. − 8 | − 9 + 4 | negative 8 vertical line negative 9 plus 4 vertical line

Algebra Solve.

13. | x | = 16 vertical line x vertical line equals 16
14. 1 = | x | 1 equals vertical line x vertical line
15. | x | + 7 = 27 vertical line x vertical line plus 7 equals 27
16. | x | − 9 = 15 vertical line x vertical line negative 9 equals 15

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