Algebra: Equations, Inequalities, Quadratics, and Translation
Core skills
Algebra questions reward translation and structure over brute force. You need: solving linear equations and inequalities; systems of two equations (substitution and elimination); quadratic techniques including factoring, the quadratic formula, the discriminant b² − 4ac, and vertex form; function notation and composition such as f(g(x)); exponent and radical laws; rational expressions; absolute value equations; and modeling word problems - distance/rate/time, combined work rates, mixture and weighted-average setups, and simple sequences.
Traps
- Dividing both sides by a variable expression: from x² = 5x, dividing by x leaves x = 5 and silently drops x = 0. Factor instead.
- Multiplying or dividing an inequality by a negative number reverses the inequality sign.
- Squaring both sides can introduce extraneous roots; always test candidates in the original equation.
- Quantity A / Quantity B comparisons require testing several numbers: 0, 1, a negative, and a fraction between 0 and 1.
- Translated phrases: 5 less than x is x − 5, not 5 − x.
- If the question fixes a range such as x > 2, report only the root that satisfies it.
Worked example
Solve x² − 5x + 6 = 0. Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. If the problem also states x > 2.5, the only valid answer is 3. Rule of thumb: move everything to one side, factor, then apply every stated constraint before answering.