Equations and Inequalities
Equations and Inequalities
This domain tests solving linear, quadratic, rational, radical, absolute-value, and systems of equations, along with linear and quadratic inequalities and interval notation. The exam emphasizes valid solution sets: a correct algebraic step can still produce an extraneous solution or an incorrectly stated interval.
Core rules
- Linear equations: isolate the variable using inverse operations. For systems, use substitution or elimination and check the ordered pair in both equations.
- Quadratics: try factoring first, then the square-root property, completing the square, or the quadratic formula x = [−b ± √(b² − 4ac)]/(2a).
- Rational equations: multiply by the LCD, solve, then reject any value that makes an original denominator zero.
- Radical equations: isolate the radical, square both sides, solve, and check every candidate in the original equation.
- Absolute value: |A| = b gives A = b or A = −b for b ≥ 0. |A| < b means −b < A < b; |A| > b means A < −b or A > b.
- Inequalities: multiplying or dividing by a negative number reverses the inequality sign. Use brackets for included endpoints and parentheses for excluded endpoints.
- Quadratic inequalities: find the critical points, test intervals on a sign chart, and include endpoints only when the inequality is ≤ or ≥.
Worked example
Solve x² − x − 6 ≥ 0.
Factor: (x − 3)(x + 2) ≥ 0. The critical points are x = 3 and x = −2. Test x < −2: both factors are negative, so the product is positive. Test −2 < x < 3: the product is negative. Test x > 3: both factors are positive, so the product is positive. Because the inequality is ≥, include the zeros. The solution is (−∞, −2] ∪ [3, ∞).
Traps that cost marks
- Forgetting to reverse the inequality sign when multiplying or dividing by a negative.
- Keeping extraneous roots from squaring both sides of a radical equation.
- Dividing both sides by a variable and losing the solution x = 0.
- Mixing up absolute-value patterns: “less than” gives a compound interval; “greater than” gives a union.
- Using parentheses at endpoints of a ≤ or ≥ inequality.
- Solving a quadratic inequality by factoring but then writing the interval between the roots without checking the sign.