Product or quotient inequality: the sign chart is enough

The idea

To solve $A(x) \times B(x) > 0$ (or $< 0$, $\geq 0$, $\leq 0$):

1. Factor if needed.
2. Sign chart with values zeroing each factor.
3. Product of signs on the last row.
4. Read the intervals that work.

Simple example

Solve $(x - 1)(x + 3) > 0$.

Roots: $x = 1$ and $x = -3$.

$x$	$-\infty$	$-3$	$1$	$+\infty$
$x - 1$	−	−	$0$	+
$x + 3$	−	$0$	+	+
Product	+	$0$	−	$0$

$S = (-\infty, -3) \cup (1, +\infty)$.

Quotient: mind the forbidden values

For $\dfrac{A(x)}{B(x)}$, we need $B(x) \neq 0$.

Same chart, but exclude the values zeroing the denominator (double bar).

Example: solve $\dfrac{x - 2}{x + 1} \leq 0$.

Roots: $x = 2$ (zeroes numerator), $x = -1$ (forbidden).

$x$	$-\infty$	$-1$	$2$	$+\infty$
$x - 2$	−	−	$0$	+
$x + 1$	−	$0$ ‖	+	+
Quotient	+	‖	−	$0$

$S = (-1, 2]$ (careful: $-1$ excluded, $2$ included).

Watch squared factors

$(x - 3)^2$ is always $\geq 0$, zero only at $3$. It never flips the overall sign.

Common mistakes

1. Not excluding forbidden values in a quotient.
2. Forgetting to include roots for large inequalities.
3. Multiplying an inequality by $(x + 1)$ without knowing its sign (often flips direction!).

?À toi de jouer

Solutions of (x + 2)(x − 5) < 0?

A.(−2, 5)B.(−∞, −2) ∪ (5, +∞)C.[−2, 5]D.{−2, 5}

Valider