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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, SAT-II, Math Kangaroo 7-8, Math Kangaroo 9-10.

The absolute value of a real number is equivalent to the two-line definition:

Being able to write the two-line definition makes many problems related to absolute value a lot more clear.

Example 1: Absolute value of a line with positive slope.

Example 2: Absolute value of a line with negative slope.

Example 3: Absolute value of a parabola (quadratic function).

A frequently used absolute value inequality is:

If a is positive then, this is equivalent to:

However, if a is negative the inequality is impossible, since an absolute value cannot be negative.

Another type of problems has the student perform the inverse operation: given a set of inequalities, turn them into a single absolute value inequality.

Symmetrize this with respect to zero:

To obtain an absolute value inequality: