Absolute Value: Review of Absolute Value

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 | 2x + 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}.