This section is a review of the material covered in the
absolute value section
of the Integers and Rational Numbers Pre-Algebra SparkNote.

The absolute value of a number a, denoted |a|, is the positive
distance between the number and zero on the number
line. It is the value of the
corresponding "unsigned" number--that is, the number with the sign
removed. The absolute value of -12, denoted |-12|, is 12. The
absolute value of 12, denoted |12|, is also 12.

To evaluate an expression which contains an absolute value, first
carry out the expression inside the absolute value sign according to
the order of operations.
Next, take the absolute value of the resulting number. Finally,
evaluate the resulting expression according to the order of
operations.

*Example 1*: What is the value of | 2*x* + 5| if *x* = - 3? *x* = 3? If *x* = - 8?

*x* = - 3: | 2(- 3) + 5| = | - 6 + 5| = | - 1| = 1

*x* = 3: | 2(3) + 5| = | 6 + 5| = | 11| = 11

*x* = - 8: | 2(- 8) + 5| = | - 16 + 5| = | - 11| = 11

In general (but not in all cases), there are 2 values of *x* which
make an equation with an absolute value true.

*Example 2*: Find the solution
set of 3| *x*| + 2 = 8 from the replacement set { -4, -2, 0, 2, 4}.

*x* = - 4: 3| - 4| + 2 = 3(4) + 2 = 14≠8. Not a solution.

*x* = - 2: 3| - 2| + 2 = 3(2) + 2 = 8. Solution.

*x* = 0: 3| 0| + 2 = 3(0) + 2 = 2≠8. Not a solution.

*x* = 2: 3| 2| + 2 = 3(2) + 2 = 8. Solution.

*x* = 4: 3| 4| + 2 = 3(4) + 2 = 14≠8. Not a solution.

The solution set is { -2, 2}.

*Example 3*: Find the solution set of 5| - 4| = 15
from the replacement set { -10, -2, 2, 6, 14}.

*x* = - 10: 5| -4| = 5| - 5 - 4| = 5| - 9| = 5(9) = 45≠15. Not a solution.

*x* = - 2: 5| -4| = 5| - 1 - 4| = 5| - 5| = 5(5) = 25≠15. Not a solution.

*x* = 2: 5| - 4| = 5| 1 - 4| = 5| - 3| = 5(3) = 15.
Solution.

*x* = 6: 5| - 4| = 5| 3 - 4| = 5| - 1| = 5(1) = 5. Not a
solution.

*x* = 14: 5| - 4| = 5| 7 - 4| = 5| 3| = 5(3) = 15.
Solution.

The solution set is {2, 14}.