Solving Linear Inequalities

One-Variable Inequalities & Number Lines

• Solve like an equation using inverse operations; the solution is a range of values.
• Graph with an open circle for < or > (the endpoint is not included) or a closed circle for ≤ or ≥ (the endpoint is included), plus a ray.
• "At least" means ≥; "at most" means ≤; "more than" and "fewer than" are strict (>, <).

Multi-Step Inequalities & the Flip Rule

• Solve the same way as multi-step equations, with one exception: multiplying or dividing both sides by a negative number flips the inequality sign.
• After every solve, check whether your last step multiplied or divided by a negative — if so, confirm the sign flipped.

Common mistake: Forgetting to flip the inequality sign when dividing or multiplying by a negative is the #1 inequality mistake.

Compound Inequalities

• "And" compound inequalities (e.g. −2 < x ≤ 5) require both conditions true at once; the graph is the overlap of both.
• "Or" compound inequalities require at least one condition true; the graph is the union of both rays.
• A three-part inequality like a < expression < b can be solved by performing the same operation on all three parts at once.

Absolute Value Inequalities

• |x| < a (a > 0) becomes −a < x < a — an "and"/between statement.
• |x| > a (a > 0) becomes x < −a or x > a — an "or" statement that splits apart.
• Isolate the absolute value first, exactly like absolute value equations, before rewriting.

Common mistake: "Less than" gives an "and" (between); "greater than" gives an "or" (splits apart) — mixing these up is the most common error here.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.