Inequalities and Absolute Value Equations Flashcards

Grade 9 · Math · Flashcards

Key rules, solution patterns, and graphing conventions for inequalities and absolute value problems.

Flashcards

Critical rule: multiplying/dividing by a negative: The inequality sign MUST be flipped (reversed) when you multiply or divide both sides by a negative number
Open circle on number line: Used for strict inequalities (< or >); means the endpoint is NOT included in the solution
Closed circle on number line: Used for inclusive inequalities (≤ or ≥); means the endpoint IS included in the solution
Solve |x| = a (a > 0): Two solutions: x = a or x = −a
Solve |x| < a (a > 0): AND inequality: −a < x < a (between)
Solve |x| > a (a > 0): OR inequality: x > a or x < −a (outside)
Absolute value with a < 0: |expression| = negative → no solution; |expression| < negative → no solution; |expression| > negative → all real numbers
Solve 3|x + 1| = 12: |x + 1| = 4 → x + 1 = 4 or x + 1 = −4 → x = 3 or x = −5
Intersection vs Union: AND: intersection (overlap). OR: union (everything in either set)
Graph of x < 2 or x ≥ 5: Open circle at 2 shaded left + closed circle at 5 shaded right (two separate rays)
Solve: 5 − 3x ≥ 14: −3x ≥ 9 → x ≤ −3 (flip sign dividing by −3)
Write compound inequality from graph: closed at −1, open at 4, shaded between: −1 ≤ x < 4
Absolute value distance interpretation: |x − a| = d means x is exactly d units away from a on the number line