Algebra: Equations and Inequalities
Core concepts
Candidates must solve linear, quadratic, rational, radical, and absolute-value equations and inequalities; use the discriminant Δ = b² − 4ac to classify quadratic roots; apply Vieta's formulas (sum −b/a, product c/a); and use sign charts for polynomial and rational inequalities. Systems of equations, including substitution and elimination, are also tested.
Traps
- Extraneous roots from squaring both sides of a radical equation or from multiplying by a variable expression. Always verify solutions in the original equation.
- Forgetting that a rational inequality's excluded zeros of the denominator cannot be solutions, even when the sign chart says "positive".
- Reversing the inequality direction when clearing a negative denominator without tracking sign cases.
- Losing the case split in |f(x)| = g(x): require g(x) ≥ 0 and solve f(x) = ±g(x).
Worked example
Solve (x − 1)/(x + 2) ≥ 0. Critical points are x = 1 (zero of numerator) and x = −2 (excluded). Testing intervals: for x < −2 the ratio is positive; for −2 < x < 1 it is negative; for x > 1 it is positive. Since the denominator cannot vanish, the solution is (−∞, −2) ∪ [1, ∞). Note the bracket at 1 (numerator zero allowed) and the parenthesis at −2.
Rule of thumb
Before dividing by an expression containing the variable, ask whether it can be zero or negative; if so, split cases or use a sign chart instead.