What is Sign table? — Mathematics

Definition 2.15 High School Mathematics · Chapter 2 — Algebra: Equations and Inequalities

A sign table records, on one row per factor, the sign ( $+$ or $-$ ) of each factor of an expression as $x$ runs over $\R$ , with a $0$ at each value where the factor vanishes; a final row gives the sign of the product, using the rule of signs column by column.

The graph of P(x) = (x-2)(3-x) confirms the sign table: positive between the roots 2 and 3, negative outside. — The graph of $P(x) = (x-2)(3-x)$ confirms the sign table: positive between the roots $2$ and $3$ , negative outside.

Examples

Example 2.16

Sign of $P(x) = (x - 2)(3 - x)$ . The factor $x - 2$ vanishes at $2$ and is positive after; the factor $3 - x$ vanishes at $3$ and is positive before:

$x$	$-\infty$	$2$		$3$	$+\infty$
$x - 2$	$-$	$0$	$+$		$+$
$3 - x$	$+$		$+$	$0$	$-$
$P(x)$	$-$	$0$	$+$	$0$	$-$

So $P(x) > 0$ exactly on $\intoo{2}{3}$ , and $P(x) \leq 0$ on $\intoc{-\infty}{2} \cup \intco{3}{+\infty}$ .

Example 2.18

Solve $\dfrac{x + 1}{x - 3} \leq 0$ . The numerator vanishes at $-1$ , the denominator at $3$ (forbidden value). The quotient is negative when the two have opposite signs, which happens for $x$ between $-1$ and $3$ . Including the bound where the numerator vanishes but excluding the forbidden value: the solution set is $\intco{-1}{3}$ .

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