Summary: Solving Single- and Multi-Step Inequalities

Key Concepts

Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign.

Symbol	Words	Example
$\neq$	not equal to	${2}\neq{8}$ , 2 is not equal to 8
$\gt$	greater than	${5}\gt{1}$ , 5 is greater than 1
$\lt$	less than	${2}\lt{11}$ , 2 is less than 11
$\geq$	greater than or equal to	${4}\geq{ 4}$ , 4 is greater than or equal to 4
$\leq$	less than or equal to	${7}\leq{9}$ , 7 is less than or equal to 9

The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.

Inequality	Words	Interval Notation
${a}\lt{x}\lt{ b}$	all real numbers between a and b, not including a and b	$\left(a,b\right)$
${x}\gt{a}$	All real numbers greater than a, but not including a	$\left(a,\infty \right)$
${x}\lt{b}$	All real numbers less than b, but not including b	$\left(-\infty ,b\right)$
${x}\ge{a}$	All real numbers greater than a, including a	$\left[a,\infty \right)$
${x}\le{b}$	All real numbers less than b, including b	$\left(-\infty ,b\right]$
${a}\le{x}\lt{ b}$	All real numbers between a and b, including a	$\left[a,b\right)$
${a}\lt{x}\le{ b}$	All real numbers between a and b, including b	$\left(a,b\right]$
${a}\le{x}\le{ b}$	All real numbers between a and b, including a and b	$\left[a,b\right]$
${x}\lt{a}\text{ or }{x}\gt{ b}$	All real numbers less than a or greater than b	$\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers	All real numbers	$\left(-\infty ,\infty \right)$

The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:

Start With	Multiply By	Final Inequality
$a>b$	$c$	$ac>bc$
$5>3$	$3$	$15>9$
$a>b$	$-c$	$-ac<-bc$
$5>3$	$-3$	$-15<-9$

The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:

Start With	Divide By	Final Inequality
$a>b$	$c$	$\displaystyle \frac{a}{c}>\frac{b}{c}$
$4>2$	$2$	$\displaystyle \frac{4}{2}>\frac{2}{2}$
$a>b$	$-c$	$\displaystyle -\frac{a}{c}<-\frac{b}{c}$
$4>2$	$-2$	$\displaystyle -\frac{4}{2}<-\frac{2}{2}$

Module 11: Linear Inequalities