Absolute Value Inequalities Calculator

Solve |ax + b| < c or |ax + b| > c.

Lower boundary -3

Upper boundary 7

Solution (-3, 7)

Formula: < gives a sandwich; > gives a union

Step-by-step with your numbers:

1. Values used:

2. a = 1

3. b = -2

4. c = 5

6. Lower boundary = -3

7. Upper boundary = 7

8. Solution = (-3, 7)

Did we solve your problem today?

Absolute-value inequalities split into two cases that give either a single interval or a union.

The math behind it

|ax+b| < c means −c < ax+b < c (a bounded interval). |ax+b| > c means ax+b > c OR ax+b < −c (a union of two rays).

|x − 2| < 5 → −3 < x < 7, i.e. (−3, 7).

What if c is negative?

For '<' there is no solution; for '>' all real numbers work.