Calculadora de Inequações de Valor Absoluto
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
|ax+b| < c ⇒ (−c−b)/a < x < (c−b)/a
About this calculator
This calculator solves absolute value inequalities in the form |ax + b| < c, where a, b, and c are real numbers and a is nonzero. It provides the interval of x values that satisfy the condition, displaying the result as x₁ < x < x₂. The tool is useful for students and professionals needing a quick and accurate solution for algebraic problems involving absolute values.
The operation is based on the definition of absolute value: |ax + b| < c means the expression ax + b is within a distance less than c from zero. This results in two simultaneous inequalities: -c < ax + b < c. Isolating x, we get x > (-c - b)/a and x < (c - b)/a, provided a > 0. If a < 0, the inequality signs are reversed. The calculator automatically considers the sign of a to present the correct interval.
Use this calculator whenever you need to solve absolute value inequalities in contexts such as error analysis, tolerance limits, or distance problems in mathematics and physics. For example, to find x values satisfying |2x - 3| < 5, simply enter a=2, b=-3, and c=5. The result will be the interval of x that meets the condition.
Caution: the calculator assumes a ≠ 0 and c > 0, since |ax + b| < c only has a solution if c > 0. If c ≤ 0, there is no real solution. Also, check that the sign of a is correct, as it affects the direction of the inequalities. This tool is not suitable for inequalities with ≥ or >.
Frequently asked questions
What if c is negative?
If c ≤ 0, the inequality |ax + b| < c has no real solution because the absolute value is always non-negative. The calculator will indicate no solution.
Does the calculator work for inequalities with ≥ or >?
No, this calculator only solves inequalities of the form |ax + b| < c. For ≥ or >, the result would be two separate intervals, and this tool does not cover those cases.
What if a is zero?
The calculator requires a ≠ 0, as the inequality would no longer be linear. If a = 0, the expression reduces to |b| < c, which is independent of x and not handled here.
How does the sign of a affect the result?
If a > 0, the interval is (-c - b)/a < x < (c - b)/a. If a < 0, isolating x reverses the inequality signs, resulting in (c - b)/a < x < (-c - b)/a. The calculator does this automatically.