Quadratic and Higher-Degree Inequalities — Cheat sheet

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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.