Enter an absolute value inequality like |ax + b| < c or |ax + b| > c by specifying the coefficients a, b, c and choosing the inequality type. The Absolute Value Inequality Solver returns the solution interval, the two boundary values, and a plain-English description of the solution set — perfect for checking your algebra homework.
Solution Interval
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Left Boundary (x₁)
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Right Boundary (x₂)
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Solution Set Type
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To solve |ax + b| < c, split it into a compound inequality: -c < ax + b < c, then solve for x by subtracting b and dividing by a from all parts. For |ax + b| > c, split into two separate inequalities: ax + b > c OR ax + b < -c, then solve each branch independently.
A 'less than' absolute value inequality (|expr| < c) produces a bounded interval — the solution is a segment between two values. A 'greater than' inequality (|expr| > c) produces an unbounded union — the solution is two rays going to negative and positive infinity. These are sometimes called 'and' inequalities and 'or' inequalities respectively.
Since absolute values are always non-negative, |ax + b| < c has no solution when c ≤ 0 (nothing can be less than a negative number). Conversely, |ax + b| > c is true for all real x when c < 0, giving a solution of all real numbers.
If your inequality is k·|ax + b| < c, divide both sides by k first. If k is positive, the inequality direction stays the same. If k is negative, the inequality flips — for example, k·|ax + b| < c becomes |ax + b| > c/k when k < 0.
Parentheses ( ) indicate a strict inequality (< or >), meaning the boundary value itself is NOT included. Square brackets [ ] indicate a non-strict inequality (≤ or ≥), meaning the boundary value IS included. For example, (-2, 5) means all x between -2 and 5, not including the endpoints.
Yes. If you have |ax + b| < 0 or |ax + b| < c where c is negative, there is no real solution because absolute values are always zero or positive. The calculator will indicate when this occurs.
If a = 0, the expression inside the absolute value reduces to |b|, a constant. The inequality then becomes a simple true/false statement: either all real numbers are solutions (if the statement is true) or there is no solution (if it is false). There is no variable x to solve for.
An absolute value equation (|ax + b| = c) has at most two specific point solutions. An absolute value inequality gives a range of solutions — either an interval between two values or a union of two rays — representing infinitely many values of x that satisfy the condition.