Inequalities, systems of inequalities and solution sets on the number line
Equivalence is everything
Two inequalities are equivalent if they have exactly the same solution set. Adding or subtracting a quantity preserves equivalence; multiplying or dividing by a negative number reverses the sign. Multiplying by an expression whose sign depends on x is never allowed - move everything to one side and use a sign chart.
The interval method: bring the inequality to f(x) ⋚ 0, factor numerator and denominator completely, mark all zeros and all points where f is undefined on the number line, then determine the sign on each interval (test a point, or use multiplicity: even multiplicity means no sign change, odd means a change). Include an endpoint only when the inequality is non-strict and the function is defined there - a zero of the denominator is never included.
A system's solution is the intersection of the individual solution sets; "and" means intersect, while a union of cases means merge.
Traps. Forgetting to flip the sign after dividing by a negative. Writing x² > 4 as x > ±2. Dropping the restriction imposed by a denominator or a square root. Confusing x² ≤ a (one segment) with x² ≥ a (two rays).
Worked example. (x − 1)(x + 2)² ≥ 0. The factor (x + 2)² is never negative, so it only forces x = −2 to be included as a zero. Otherwise the sign follows (x − 1): negative for x < 1, non-negative for x ≥ 1. Solution: x ∈ {−2} ∪ [1, +∞).