Number Line & Absolute Value

Absolute value: |x| = distance from zero. Always non-negative.

|5| = 5. |-5| = 5. |0| = 0.

Equations: |x - 3| = 7

Case 1: x - 3 = 7, so x = 10

Case 2: x - 3 = -7, so x = -4

Inequalities:

|x| < 5 means -5 < x < 5 (between)

|x| > 5 means x < -5 OR x > 5 (outside)

Key insight: |a - b| = the distance between a and b on the number line. This interpretation solves many GRE problems faster.

Quantitative Comparison trap:

Quantity A: |x + y|

Quantity B: |x| + |y|

B is ALWAYS greater than or equal to A (triangle inequality). Equal when x and y have the same sign. A < B when they have different signs.

Reference:

Wikipedia: Absolute Value

https://en.wikipedia.org/wiki/Absolute_value