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GMAT Quant
Absolute Value Basics
Interpret absolute value as distance and solve equations by accounting for both directions.
Meaning
Absolute value is distance on a number line.
The expression |x - a| is the distance between x and a, so it can never be negative.
Distance ignores direction.
- Expression
- |9 - 4|
- Distance
- 5
- Reverse
- |4 - 9|
- Distance again
- 5
Method
An equation |A| = B usually creates two cases.
When B is nonnegative, A can equal B or -B. The two cases represent points the same distance from zero.
Example: |x - 6| = 4
- Case 1
- x - 6 = 4 -> x = 10
- Case 2
- x - 6 = -4 -> x = 2
- Solutions
- 2 and 10
Recognition
A negative right side is impossible.
Because absolute value is never negative, an equation such as |2x + 1| = -3 has no solution.
Recognition
Use this recognition pattern.
- 01Absolute value around a variable often means distance.
- 02An equality can create two symmetric cases.
- 03A zero right side creates one case: the inside must equal zero.
- 04A negative right side means no solution.
Mini application
Try this: |3x + 2| = 8
Find all possible values of x before scrolling.
Solution
The solutions are x = 2 and x = -10/3.
Solve 3x + 2 = 8 and 3x + 2 = -8 separately.
Remember this
Distance creates symmetry.
For |A| = B, think "B units to the right or B units to the left" before doing algebra.