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Solving Inequalities

Solving Inequalities by Using Addition and Subtraction

  For all numbers a, b, and c, the following are true.
Addition and Subtraction Properties of Inequalities
  1. If a > b, then a + c > b + c and a - c > b - c. (Also true for )
  2. If a < b, then a + c < b + c and a - c < b - c. (Also true for )

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.

  For all numbers a, b, and c, the following are true.
Multiplication and Division Properties for Inequalities
  1. If c is positive and a < b, then ac < bc and , and if c is positive and a b, then ac > bc and .
  2. If c is negative and a < b, then ac > bc and , and if c is negative and a > b, then ac < bc and .

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} .

 
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