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Linear Inequality Calculator with Steps and Interval Notation

Solve a linear inequality such as 2x − 5 < 3x + 1 step by step, flipping the sign when dividing by a negative, and get interval notation, set-builder notation and the number line.

Inequality
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How it works

Same steps as an equation (add, subtract, multiply, divide both sides) — but multiplying or dividing both sides by a negative number reverses the inequality sign

Linear inequalities are solved like linear equations, with one extra rule: whenever you multiply or divide both sides by a negative number the direction of the sign flips (< becomes >, ≤ becomes ≥). The solution is a whole range of numbers, written three ways: as an inequality (x > −6), in interval notation ((−6, ∞), where a round bracket excludes the endpoint and a square bracket includes it) and on a number line (open circle for < or >, closed circle for ≤ or ≥).

Worked example

Solve 2x − 5 < 3x + 1. Subtract 3x from both sides: −x − 5 < 1. Add 5 to both sides: −x < 6. Multiply both sides by −1 and reverse the sign: x > −6. In interval notation this is (−6, ∞); on a number line, an open circle at −6 shaded to the right.

Sources: Solving inequalities — Math is Fun · Linear inequality — Wikipedia

Frequently asked questions

Why does the sign flip when dividing by a negative?

Multiplying by a negative reverses order on the number line: 2 < 5, but −2 > −5. To keep the statement true, the inequality sign must turn around.

What is the difference between ( and [ in interval notation?

A parenthesis means the endpoint is not included (strict < or >); a bracket means it is included (≤ or ≥). Infinity always gets a parenthesis.

How do I type ≤ and ≥?

Type <= or >= and the calculator converts them. You can also paste the symbols ≤ and ≥ directly.

Can it solve compound inequalities like −3 < 2x + 1 < 7?

Solve each part separately (−3 < 2x + 1 and 2x + 1 < 7) and keep the overlap. For absolute value inequalities, use the absolute value equation calculator, which handles them directly.

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This calculator is for educational purposes. Double-check important results before acting on them.