Absolute Value Inequalities
Solve absolute value inequalities of the form |ax + b| < c or |ax + b| > c with step-by-step solutions. Free online absolute value inequality solver.
About This Calculator
This calculator solves absolute value inequalities of the form |ax + b| < c, |ax + b| <= c, |ax + b| > c, or |ax + b| >= c. The solution is expressed as an interval (for < and <=) or a union of two intervals (for > and >=).
For < inequalities, the solution is a bounded interval between two values. For > inequalities, the solution is two disjoint intervals extending to infinity. This calculator handles all cases including c = 0 and c < 0.
Frequently Asked Questions
How do you solve |ax + b| < c?
For |ax + b| < c (with c > 0), solve -c < ax + b < c. This gives a bounded interval: (-c - b)/a < x < (c - b)/a.
How do you solve |ax + b| > c?
For |ax + b| > c (with c >= 0), solve ax + b < -c or ax + b > c. This gives two disjoint intervals: x is outside the range [-c, c] shifted by b and scaled by a.
What happens if c is negative?
If c < 0: for |x| < c, there is no solution (absolute value is always >= 0). For |x| > c, the solution is all real numbers (since |x| >= 0 > c for all x).
What is the difference between |x| < c and |x| > c?
|x| < c means the distance from x to 0 is less than c, so x is between -c and c (bounded interval). |x| > c means the distance is greater than c, so x is outside that interval (two unbounded regions).
Where are absolute value inequalities used?
They model tolerance ranges in manufacturing (|x - target| <= tolerance), error bounds in scientific measurements, confidence intervals in statistics, and range constraints in optimization problems.
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration required.