Expressions, Equations, and Inequalities: Linear Relationships
What this domain tests
This domain covers simplifying linear expressions, solving two-step equations and inequalities, and translating verbal conditions into symbols. FAST items may ask for the solution, the graph of a solution set, or the equation that models a situation.
Core rules
- Simplify by distributing first, then combining like terms: 3(x + 4) − 2x = 3x + 12 − 2x = x + 12.
- Solve an equation by undoing operations in reverse order: undo addition or subtraction, then multiplication or division. Keep both sides balanced.
- Check a solution by substituting it back into the original equation.
- Solve an inequality the same way, with one exception: when you multiply or divide both sides by a negative number, reverse the inequality symbol.
- Graph inequalities on a number line with an open circle for < or > and a closed circle for ≤ or ≥. Shade in the direction that makes the inequality true.
- Translate carefully: “at least” means ≥, “at most” means ≤, “no more than” means ≤, and “more than” means >.
Worked example
Solve −2x + 5 ≥ 11 and graph the solution. Subtract 5 from both sides: −2x ≥ 6. Divide both sides by −2 and reverse the symbol: x ≤ −3. The graph has a closed circle at −3 and is shaded to the left. Check a value such as x = −4: −2(−4) + 5 = 13, and 13 ≥ 11 is true.
Traps
- Forgetting to reverse the inequality when dividing by a negative, which gives x ≥ −3 instead of x ≤ −3.
- Distributing a negative incorrectly: −2(x − 3) = −2x + 6, not −2x − 6.
- Using an open circle for ≤ or ≥.
- Dividing only one side of an equation by a coefficient.
- Treating “at least 20” as x > 20 rather than x ≥ 20.