Solving linear inequality

Solving an inequality means isolating the variable and then graphing the solution on a number line, or writing it in interval or set-builder notation. To solve an inequality, you use the following properties of inequality.

Addition and subtraction properties of inequality

For any real numbers $a$, $b$, and $c$,

if $a\lt b$, then

$a+c\lt b+c$

and

$a-c\lt b-c$

Multiplication and division properties of inequality

For any real numbers $a$, $b$, and $c$,

if $a\lt b$, then

$ac\lt bc$ if $c$ is positive.

$ac\gt bc$ if $c$ is negative.

and

$\dfrac{a}{c}\lt \dfrac{b}{c}$ if $c$ is positive.

$\dfrac{a}{c}\gt \dfrac{b}{c}$ if $c$ is negative.

Note that if you multiply or divide by a negative number, the inequality sign reverses. That is, $\lt$ becomes $\gt$, and vice versa.

If you switch sides — moving everything on the right to the left and everything on the left to the right — you must also reverse the inequality sign. For example,

$4\lt x \:$ is same as $\: x \gt 4$

$y\ge -4 \:$ is same as $\: -4 \le y$

Compound inequalities →