Expressions, Equations and Inequalities
This domain tests algebraic fluency: simplifying expressions, solving linear equations and inequalities, working with systems, and using square and cube roots. Questions often combine two skills, such as solving an equation and interpreting the solution in context.
Expressions and linear equations
Simplify by distributing and combining like terms; a negative sign in front of parentheses changes every sign inside. Solve multi-step equations by undoing operations in reverse order, and keep the equation balanced. For a linear relationship y = mx + b, m is the rate of change (slope) and b is the initial value (y-intercept). Slope from two points is (y₂−y₁)/(x₂−x₁); keep the subtraction order consistent.
Inequalities
Solve like equations, but reverse the inequality symbol when you multiply or divide both sides by a negative number. Graph solutions on a number line, using an open circle for < or > and a closed circle for ≤ or ≥. Interpret phrases: “at least” means ≥, “at most” means ≤, and “no more than” means ≤.
Systems of equations
A system’s solution is the ordered pair that satisfies both equations. Substitution works well when one variable is isolated. Elimination works well when coefficients are opposites or can be made opposites. If the equations describe parallel lines, there is no solution; if they are equivalent, there are infinitely many.
Radicals
√a asks for the nonnegative number whose square is a; ∛a asks for the number whose cube is a. Solve x² = p: if p > 0, x = ±√p; if p = 0, x = 0; if p < 0, there is no real solution. Solve x³ = p with one real solution, x = ∛p. Squaring both sides of an equation can introduce extraneous solutions, so check answers.
Worked example
Solve the system x + y = 10 and 2x − y = 2. Adding the equations eliminates y: 3x = 12, so x = 4. Substitute into x + y = 10: 4 + y = 10, so y = 6. The solution is (4, 6). Check in the second equation: 2(4) − 6 = 2, which is true.
Traps that cost marks
- Forgetting to flip the inequality when dividing by a negative.
- Distributing a negative to only the first term inside parentheses.
- Accepting a system answer that satisfies one equation but not the other.
- Writing only the positive root of x² = 49 when both roots are required.
- Reversing slope subtraction order between numerator and denominator.