A typical literal inequality

A literal inequality contains, besides the unknown $x$, a parameter (for example $a$). To solve it means to discuss the solution as the parameter varies.

Example — A typical literal inequality

Solve $x(ax-2)\le 0$ as $a\in\mathbb{R}$ varies.

The factors are $f_1: y=x$ and $f_2: y=ax-2$. The sign of $f_2$ depends crucially on the value of $a$: three cases must be distinguished.

Case $a=0$: $f_2 = -2 < 0$ always. The inequality becomes $-2x\le 0$, that is $x\ge 0$. Solution: $\boxed{x\ge 0}$.

Case $a > 0$: $f_2 = ax-2$ is an increasing straight line with a zero at $x=\frac{2}{a} > 0$.

	$x<0$	$x=0$	$0<x<\frac{2}{a}$	$x=\frac{2}{a}$	$x>\frac{2}{a}$
$f_1=x$	$-$	$0$	$+$	$+$	$+$
$f_2=ax-2$	$-$	$-$	$-$	$0$	$+$
$f_1\cdot f_2$	$+$	$0$	$-$	$0$	$+$

Solution: $\boxed{0\le x \le \dfrac{2}{a}}$.

Case $a < 0$: $f_2 = ax-2$ is a decreasing straight line with a zero at $x=\frac{2}{a} < 0$.

	$x<\frac{2}{a}$	$x=\frac{2}{a}$	$\frac{2}{a}<x<0$	$x=0$	$x>0$
$f_1=x$	$-$	$-$	$-$	$0$	$+$
$f_2=ax-2$	$+$	$0$	$-$	$-$	$-$
$f_1\cdot f_2$	$-$	$0$	$+$	$0$	$-$

Solution: $\boxed{x \le \dfrac{2}{a} \;\text{oppure}\; x\ge 0}$.

Summary: the sign of the parameter changes not only the position of the key points but also the slope of the factor $f_2$, and hence the way the signs are distributed in the table.

Links

Topics: Inequalities
Concepts: Key point · Literal inequality · Sign analysis · Sign table
Skills: Reasoning by cases · Solving inequalities