Linear Equations & Inequalities

Solving Linear Inequalities

Solving and graphing a linear inequality in one variable on a number line.

Intermediate20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Multi-Step Equations

When the Answer Is a Range, Not a Point

An equation like x + 3 = 7 has exactly one solution. An inequality like x + 3 < 7 has infinitely many — every number less than 4 makes it true. Solving a linear inequality uses almost the same steps as solving an equation, with one rule that has no equation counterpart: multiplying or dividing by a negative number reverses the direction of the inequality.

To see why, consider the true statement 2 < 5. Multiplying both sides by −1 gives −2 and −5 — but −2 is greater than −5, not less. The direction has to flip to keep the statement true: −2 > −5.

Worked Example — A Two-Step Inequality

Solve 3x − 4 < 11. Add 4 to both sides: 3x < 15. Divide both sides by 3 (a positive number, so the inequality direction stays the same): x < 5.

Worked Example — Dividing by a Negative Flips the Inequality

Solve −2x + 6 ≥ 10. Subtract 6 from both sides: −2x ≥ 4. Divide both sides by −2, and flip the inequality direction because you're dividing by a negative: x ≤ −2.

The solution to an inequality is graphed on a number line: an open circle at the boundary value for < or >, meaning that value itself is not included, and a closed (filled-in) circle for ≤ or ≥, meaning it is included. An arrow extends from the circle in the direction of every solution.

Tip

Whenever you divide or multiply an inequality by a negative number, flip the inequality symbol at that exact step — not before, and not after — so it's easy to double check later.

Common Mistakes

Forgetting to flip the inequality symbol when dividing or multiplying by a negative number.
Every time a negative number multiplies or divides both sides, the inequality direction reverses. Circle or note that step specifically while solving, until it becomes automatic.
Using an open circle for ≤ or ≥, or a closed circle for < or >, when graphing the solution.
A closed (filled) circle means the boundary value is included in the solution — that only happens with ≤ or ≥. An open circle means it is excluded, which happens with strict < or >.

Key Takeaways

Solving a linear inequality uses the same inverse operations as solving an equation.
Multiplying or dividing both sides by a negative number reverses the inequality's direction.
On a number line, use an open circle for < or >, a closed circle for ≤ or ≥, and an arrow showing every solution.

Summary

A single linear inequality describes a whole range of solutions on one side of a boundary value. Next, you'll combine two inequalities at once to describe more precise ranges.

Track your progress

← Literal Equations Back to Linear Equations & Inequalities Compound Inequalities→