Absolute Value Greater-Than Inequality Solver

Because a greater-than absolute value condition describes values outside a central range, the answer usually consists of two separate rays. The result includes standard interval notation, structured interval endpoints, boundary inclusion details, and classifications for special cases, making it useful for learning, checking work, or feeding a deterministic answer into another application.

Why a greater-than absolute value inequality creates two rays

Absolute value measures distance from zero, so an expression such as |2x - 6| describes how far the value of 2x - 6 lies from zero. Requiring that distance to be greater than 4 means the expression must lie either below -4 or above 4. Those alternatives point in opposite directions and produce two separate ranges of x rather than one continuous interval. For |2x - 6| > 4, the two branches are 2x - 6 < -4 and 2x - 6 > 4. Solving them gives x < 1 or x > 5, written as (-∞, 1) ∪ (5, ∞). The union symbol matters because both rays belong to the answer even though the values between the boundary points do not. This solver performs that split consistently and returns each ray as a structured interval as well as a readable solution set, which helps prevent the common mistake of joining the endpoints into a single interval or keeping only one branch.

Enter the inequality and read its boundaries

Enter one linear expression in x between absolute-value bars, followed by > or >= and a numeric constant. Forms such as |x + 3| > 7, |-0.5x + 2| >= 4, and |3*x - 1| > 0 are accepted. The multiplication symbol before x is optional, ordinary decimal coefficients are supported, and the Unicode greater-than-or-equal sign may be used. The response identifies the two boundary points in increasing order and states whether each finite endpoint is included. Strict > produces parentheses because equality at the boundary does not satisfy the original statement. Inclusive >= produces square brackets at finite endpoints because boundary equality is allowed. Infinity always uses a parenthesis. The structured intervals also expose left and right endpoints and closure flags, so software can consume the result without parsing the displayed notation. Expressions must remain linear: powers, multiple variables, fractions written with a slash, and a variable on the right side are outside this capability’s deliberately narrow and predictable contract.

Understand special cases and verify the result

Most valid inputs return two rays, but a complete solver must also handle cases in which that geometry changes. If the right side is negative, every real x satisfies a nonnegative absolute value being greater than that negative number. If the comparison is >= 0, every real number also qualifies. With a strict comparison against zero, the single root of the inner linear expression is excluded, which appears naturally as two rays meeting at an omitted point. A zero x coefficient makes the expression constant; the solver evaluates the comparison directly and returns either every real number or the empty set. To check an ordinary two-ray result, substitute a value from the excluded middle region and one value from each outer ray into the original inequality. The middle test should fail while both outer tests should pass. This quick substitution confirms the direction of the rays, and testing the exact boundary points confirms whether parentheses or brackets are correct.