## 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 $a>b$ $c$ $\displaystyle \frac{a}{c}>\frac{b}{c}$ $4>2$ $2$ $\displaystyle \frac{4}{2}>\frac{2}{2}$ $a>b$ $-c$ $\displaystyle -\frac{a}{c}<-\frac{b}{c}$ $4>2$ $-2$ $\displaystyle -\frac{4}{2}<-\frac{2}{2}$