Compound Inequalities

Algebra 1 · Axiom Academy

EXAMPLE Solving Compound Inequalities Work an "AND" inequality by splitting it into two pieces, solving each, and taking the overlap. Solve the compound inequality and describe its solution set. Nice work. You solved a three-part "AND" inequality by working each piece and keeping only the numbers that satisfy both. Split: A compact inequality like is two inequalities at once — keep the middle expression 2x - 1 in both halves, joined by "AND". Solve each half: Use the same algebra on both: add 1, then divide by 2. Dividing by the positive number 2 keeps every inequality sign pointing the same way. Intersect: "AND" means BOTH must hold, so the answer is the overlap of x > 2 and . Read the endpoints: The strict < leaves 2 out (open circle); the includes 4 (closed circle). So the solution is . The shaded band is every x with : open at 2 (not included), closed at 4 (included). Every "AND" inequality follows this rhythm: split, solve, intersect, then read the endpoints carefully.

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