Linear Equations and Inequalities

ACT Math · Axiom Academy

LESSON Linear Equations and Inequalities Solve multi-step equations like a balance staying level, then carry that same logic to inequalities — with one crucial twist when a negative number gets involved. 1. An Equation Is a Balance That Must Stay Level Picture 3x + 5 = 2x + 12 as a two-pan scale: the left pan holds 3x+5 , the right pan holds 2x+12 , and the scale is level because both sides are equal. Any move you make — subtracting 2x , subtracting 5 — has to happen on both pans at once , or the scale tips and the equation breaks. Same move, both pans — still level Watch it work: 2(x+3) = 5x - 9 Distribute first: 2x + 6 = 5x - 9 . Subtract 2x from both pans: 6 = 3x - 9 . Add 9 to both pans: 15 = 3x . Divide both pans by 3 : Fractions clear the same way — multiply the whole scale by the LCD. To solve , multiply every term (both pans) by 6 : 2x + 2x = 30 , so 4x = 30 , giving 2. Multiplying by a Negative Reverses the Order Inequalities solve exactly like equations — distribute, combine, isolate — with one exception you can actually see rather than memorize: take two numbers where , and multiply both by -3 . Now -6 sits to the right of -15 on the number line, so — the order reversed. That reversal is the entire sign-flip rule. Two ordered points, multiplied by a negative Order reverses — the sign must flip to stay true Subtract 4 , divide by +2 (no flip): Subtract 5 , divide by -2 — FLIP:

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