Absolute Value Inequality Solver

How it is calculated

The tool solves |ax + b| ⋄ c, where ⋄ is one of <, ≤, >, ≥ (codes lt, le, gt, ge). It finds the two boundary points where the inside equals −c and +c: x₁ = (−c − b) ÷ a and x₂ = (c − b) ÷ a, then sorts them into a lower and an upper bound, so a negative a still gives them in the right order. The "less than" types give one interval between the bounds; the "greater than" types give the two outer rays. Strict signs exclude the endpoints, ≤ and ≥ include them. With a = 0 or an empty field there is no result. If c is negative, |…| < c and |…| ≤ c have no solution and |…| > c and |…| ≥ c hold for every real number. With c = 0, < has no solution, ≤ gives the single point x = −b ÷ a, ≥ gives all reals, and > gives every x except that point. Bounds are printed to 4 decimals.

Common mistakes

• Choosing the wrong direction. "Less than" gives one interval between the bounds, while "greater than" gives two outer pieces; mixing them up flips the whole answer.
• Entering b with the wrong sign. For |2x − 3| the constant is b = −3, not 3, which would move both bounds by 3 ÷ 2.
• Writing the inequality as |ax + b| > c when it is really |ax − b| > c, or forgetting that the coefficient of x belongs in a. Rewrite the expression as ax + b first, for example |3 − x| becomes a = −1, b = 3.
• Expecting a range at c = 0. For ≤ the result is the single point x = −b ÷ a, and for > it is every x except that point, which is displayed as x < p or x > p with the same number twice.
• Treating a negative c like a normal bound. An absolute value can never be below a negative number, so that case is empty by definition, and "greater than" is true for every real number.