Algebraic Expressions, Equations, and Inequalities
What this domain tests
Algebraic expressions, equations, and inequalities require manipulating symbols accurately and knowing which operations preserve equivalence. Expect linear equations, systems, quadratic factoring, exponent rules, and inequalities whose direction can change.
Core rules
- Distribute completely: -3(x - 4) = -3x + 12.
- Combine only like terms; 3x + 2x^2 cannot be simplified to 5x^2.
- For a linear equation, isolate the variable by reversing the order of operations.
- For a system, substitution or elimination works; elimination is fastest when coefficients are opposites.
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
- For quadratics, factor when possible and set each factor equal to zero; check apparent solutions in the original equation.
Worked example
Solve 3(x - 2) + 4 = 2x + 7. Distribute: 3x - 6 + 4 = 3x - 2. So 3x - 2 = 2x + 7. Subtract 2x: x - 2 = 7. Add 2: x = 9. Check: left side 3(7) + 4 = 25; right side 18 + 7 = 25. The solution is consistent.
Traps
The most common error is distributing a negative incorrectly: -(x - 5) = -x + 5, not -x - 5. Do not cancel a term that is added rather than multiplied. When solving an inequality such as -2x + 5 < 11, subtracting 5 gives -2x < 6, and dividing by -2 gives x > -3; the sign must flip. Squaring both sides can introduce extraneous roots, so substitute back.