Equations and Inequalities
Core concepts
Solve linear, quadratic, rational, radical, exponential, logarithmic, and absolute-value equations, plus systems (substitution, elimination). For quadratics, use factoring, completing the square, or the quadratic formula; the discriminant b² − 4ac tells you the nature of the roots. For inequalities, solve as equalities first, then test intervals.
Traps
- Multiplying or dividing an inequality by a negative flips the direction.
- Absolute value: |x| < a means −a < x < a; |x| > a splits into two rays.
- Rational inequalities: never multiply by a variable expression without checking sign; use a sign chart with critical points.
- Quadratic inequalities: the parabola's opening direction determines whether the solution is between or outside the roots.
- Systems: check that a solution satisfies every equation, not just one.
Rule of thumb
For any inequality, move everything to one side, factor, find critical points, and test a sample point in each interval.
Worked example
Solve x² − x − 6 > 0. Factor: (x − 3)(x + 2) > 0. Critical points x = −2 and x = 3. Testing x = 0 gives −6 < 0, so the middle interval fails. Testing x = −3 gives 6 > 0 and x = 4 gives 6 > 0. Solution: x < −2 or x > 3, i.e. (−∞, −2) ∪ (3, ∞).