Overview
A linear inequality resembles a linear equation but uses a comparison symbol instead of an equals sign. Its solution is not one number but a set of numbers, which can be written in interval notation or drawn on a number line.
Formula
To solve ax + b ⋚ c, subtract b and divide by a. If a is negative, reverse the inequality symbol.
How it works
Isolate the x term
Subtract b from both sides.
Divide by a
Keep the symbol if a is positive, and reverse it if a is negative.
Write the solution
Express it as x ⋚ value.
Write the interval
Use the correct bracket for the boundary and the solution direction.
Worked examples
Example 1: 2x + 3 < 11
2x < 8, so x < 4, with interval (−∞, 4).
Example 2: −2x + 1 ≥ 5
−2x ≥ 4. Divide by −2 and flip: x ≤ −2, with interval (−∞, −2].
Frequently asked questions
Why does the sign flip when dividing by a negative?
Multiplying or dividing by a negative number reverses the order of numbers on the number line.
What is the difference between < and ≤?
< excludes the boundary, while ≤ includes it in the solution.
How do I write the solution in interval notation?
Use (−∞, k) or (k, ∞) for excluded boundaries, and [ or ] when the boundary is included.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.