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Absolute value

Absolute Value Inequalities

The distance interpretation for absolute value provides a good basis for solving some inequalities that involve absolute value. Consider the following examples.

Solve |x|<2, and graph the solution set.

Solution:
The number, x, must be less than 2 units away from zero. Thus |x|<2 is equivalent to

x>-2 and x<2

The solution set is (-2,2), and its graph:

Solve |x+3|<1, and graph the solutions.

Solution:
The number, x+3, must be less than 1 unit away from zero. Thus |x+3|<1 is equivalent to

x+3>-1 and x+3<1

Solving each equation of the disjunction yields

 x+3>-1 x>-4 or or x+3<1 x<-2

The solution set is (-4,-2) and its graph:

The following general property should seem reasonable.

|ax+b|<k is equivalent to ax+b>-k and ax+b<k, where k is a positive number.