Inequalities
Cheat sheet.

Key method

At a zero of odd multiplicity the sign changes; at even multiplicity it does not. Include zeros for $\le$ or $\ge$, and exclude them for strict inequalities. The zero-product rule alone does not solve an inequality.

Example

Solve $(x-1)(x+2)\le0$. The zeros divide the line at $-2$ and $1$. At $x=-3$, both factors are negative and the product is positive. At $x=0$, the product is negative. At $x=2$, both factors are positive. Include the endpoints, giving $[-2,1]$.

Avoid this mistake

Multiplying or dividing an inequality by a negative number reverses its direction. Dividing by an expression of unknown sign requires cases.

Key method

Move all terms to one side, form one fraction, factor numerator and denominator, and test intervals separated by every critical point. Include eligible numerator zeros for non-strict inequalities; never include denominator zeros.

Example

Solve $(x-1)/(x+2)\ge0$. Exclude $-2$, and split at $-2,1$. Testing $-3,0,2$ gives positive, negative, positive. Thus

At $1$ the expression is zero; at $-2$ it is undefined.

Avoid this mistake

Cross-multiplying by $x+2$ without knowing its sign can reverse the inequality in part of the domain and produce the wrong answer.

Key method

Draw intervals on the same number line and retain only the overlap. The endpoint is included only if all required conditions include it.

Example

Solve $x^2\le9$ and $x>1$. The first gives $[-3,3]$, the second $(1,\infty)$. Their intersection is $(1,3]$. For $\sqrt{x+3}/(x-2)$, combine $x\ge-3$ with $x\ne2$: the domain is $[-3,2)\cup(2,\infty)$.

Avoid this mistake

Adding solution intervals together by union solves an “or” question, not an “and” question. Some intersections are empty.