# Solving Linear Inequalities

Inequalities are mathematical sentences comparing two quantities that are not equal (or possibly not equal). There are five inequality symbols:

 $x\ne 3$ $x$ is not equal to $3$ $x<3$ $x$ is less than $3$ $x>3$ $x$ is greater than $3$ $x\le 3$ $x$ is less than or equal to $3$ $x\ge 3$ $x$ is greater than or equal to $3$

Often, for the last four types of inequalities, we need to solve the inequality so that the variable is alone on one side. This is done using analogues of the properties of equality : adding or subtracting the same quantity to both sides, or multiplying or dividing both sides by the same quantity. The only important difference is that:

• Whenever you multiply or divide both sides of an inequality by a negative number , you need to reverse the direction of the inequality.

To see why, consider a simple inequality like $1<2$ . If we multiply both sides by $-1$ without changing the sign, we get

$-1<-2$ , which is clearly false!

Example:

Solve for $x$ .

$-3x+2\le 14$

First, subtract $2$ from both sides.

$-3x\le 12$

Now divide both sides by $-3$ . Remember to reverse the inequality.

$x\ge -4$

Once you have the solution you might be asked to graph it .