Solving Linear Equations and Inequalities

GRE Quantitative · Axiom Academy

LESSON Solving Linear Equations and Inequalities Isolate the variable with inverse operations — and learn the one rule that trips up almost everyone: inequalities flip their sign. 1. Isolating x With Inverse Operations A linear equation keeps x to the first power only. To solve it, undo every operation attached to x — in reverse order — applying the same move to both sides so the equation stays balanced. Undo addition/subtraction first Then undo multiplication/division 2. The Flip Rule — Why Direction Reverses Inequalities use , , , instead of = , and every addition/subtraction move works exactly like an equation. But multiplying or dividing by a negative number reverses which number is bigger — so the inequality sign must flip to stay true. , and adding 10 to both sides keeps it true: . Shifting both sides never changes their order. , but multiply both sides by -1 : . The bigger number became the smaller one — the sign must flip to match. Scaling both sides by a positive number preserves order — no flip needed, same as with equations. Flip the inequality sign exactly when you multiply or divide both sides by a negative number. That's the only exception. Subtract 2 from both sides (no flip): . Now divide both sides by -3 — dividing by a negative , so the sign flips: . Check x = -4 : ✓. Check x = -2 (outside the solution since ): -3(-2)+2 = 8 , which is not greater than 11 — confirming the flipped direction is correct. 3. Extending It — Compound and Absolute-Value Inequalities

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