Linear Inequality Calculator

Enter ax + b <, ≤, >, or ≥ c to isolate x and show the sign-reversal rule when dividing by a negative coefficient.

Stable numericsSubstitute / reconstructIndependent checkLocal calculation

Relation

—Solve ax + b relation c

—Boundary

—Sign reversal

—x coefficient

—Final relation

Linear Inequality Calculator: Steps, Interval & Graph

Solve a linear inequality with sign-flip evidence, interval notation, boundary verification, worked steps and a number-line graph.

Display precision

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Independent algebra verification

The primary calculation is paired with a substitution, invariant, inverse operation, reconstruction, or equivalent-form check so the displayed answer is easier to audit.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.

Practical guide and verification

The sign flips only when multiplying or dividing by a negative value

Adding or subtracting the same number from both sides preserves the inequality direction. The reversal rule applies when the operation changes order by using a negative multiplier or divisor.

Use interval notation and the number line together

An open endpoint corresponds to a strict < or > relation, while a closed endpoint corresponds to <= or >=. The shaded direction should agree with the solved relation, giving a visual check on the algebra.

Test one point from the shaded region

Choose a simple value that the number line says should satisfy the inequality and substitute it into the original expression. Also test a value on the opposite side when practical. This verifies both the boundary and direction.

Handle a zero x coefficient as a special case

If the x terms cancel, the remaining numeric statement can be always true or always false. There is then no single finite boundary to graph as though it were an ordinary half-line.

Do not extend a linear solver to nonlinear restrictions

Products, denominators containing x, absolute values, powers, and other nonlinear terms can create multiple intervals or excluded points. Use a solver designed for that structure rather than simplifying it into an unsupported linear form.