Algebra: Expressions, Equations, Inequalities, and Functions
What you must know
Algebra questions test manipulating expressions, solving linear and quadratic equations, systems of equations, inequalities, and functions. Be fluent with factoring, the distributive property, exponent rules, and the quadratic patterns (difference of squares, perfect squares). Know that a system has a unique solution when the equations are independent and consistent.
Traps
- Sign errors when distributing a negative across parentheses.
- Dividing by a variable: dividing both sides by x discards the solution x = 0. Factor instead of dividing.
- Inequality flip: multiplying or dividing by a negative reverses the inequality sign.
- Extraneous roots: squaring both sides can introduce solutions that fail the original equation; always check.
- Function notation: f(x + 1) means substitute (x + 1) everywhere x appears, not f(x) + 1.
Worked example
Solve x² − 5x + 6 = 0 by factoring: (x − 2)(x − 3) = 0, so x = 2 or x = 3. If instead the problem asks for the sum of solutions, use the coefficient relationship: sum = −b/a = 5, product = c/a = 6. Recognizing these shortcuts saves time and reduces arithmetic slips.
Rule of thumb
When a variable appears in every term, factor it out rather than divide. Before finalizing, substitute candidate solutions back into the original equation - especially after squaring or cross-multiplying.