Solving Inequalities

Solving Inequalities by Using Addition and Subtraction

The solutions of an inequality can be graphed on a number line or writtenusing set-builder notation

Example

Solve 3 m - 7 > 4 m + 1. Check your solution, and graph it on a number line.

Solution

3m - 7 > 4m + 1

3m - 7 - 3m > 4m + 1 - 3m

- 7 > m + 1

- 7 - 1 > m + 1 - 1

- 8 > m or m < -8

In set builder notation, the solution set is {m | m < -8, which is read "the set of all numbers m such that m is less than -8". Only numbers less than - 8 substituted into the original inequality should yield a true statement.

Since only the number less than - 8 yields a true statement, the solution checks.Graph the point - 8 using an open circle, since - 8 is not part of the solution.Then draw a heavy arrow to the left to indicate numbers less than - 8.

Solving Inequalities by Using Multiplication and Division

When you multiply or divide each side of an inequality by a negative number, you must reverse the direction of the inequality symbol.

These properties also hold true for inequalities involving and .

Example

Solve -5 y 12 and check your solution.

Solution

-5 y 12

Divide each side by -5 and change the to .

Check: Let y be  2.4 and any number greater than -2.4, such as 0.

-5(-2.4) 12	-5(0) 12
12 12	0 12

In set builder notation, the solution set is { y | y 2.4} .