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 Number of inequalities to solve: 23456789
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# Solving Inequalities with Fractions and Parentheses

After studying this lesson, you will be able to:

• Solve inequalities with fractions and parentheses.

Below are the steps for solving inequalities. Remember we are applying the same rules as we did for equations. If an inequality contains fractions, the fractions can be cleared out by multiplying every term in the inequality by the common denominator. Also, if an inequality contains parentheses, the parentheses can be removed by using the distributive property.

If we multiply or divide an inequality by a negative, we reverse the inequality symbol.

The steps for solving inequalities are the same as those for solving equations:

1. Remove parentheses and clear fractions (if necessary)

2. Collect like terms on each side of the inequality symbol

3. Get the variables together on one side

4. Isolate the variable

5. Check

Example 1

 -6 < 5y - (2y - 9) We need to distribute to remove the parentheses Since there is a negative in front of the parentheses, we distribute a negative 1. -6 < 5y -2y + 9 We get this after distributing a -1 -6 < 3y + 9 Add like terms -6 - 9 < 3y + 9 - 9 Subtract 9 from each side -15 < 3y Divide each side by 3 -5 < y

Check by substituting into the original inequality

Example 2

 3 (3x + 6 ) < 3 ( 6x - 9 ) Remove parentheses by multiplying 9x + 18 < 18x - 27 Now, we need to get the variables together 9x + 18 - 9x < 18x - 27 - 9x Subtract 9x from each side 18 < 9x - 27 18 + 27 < 9x - 27 + 27 Add 27 to each side 45 < 9x Divide each side by 9 5 < x

Check by substituting into the original inequality

Example 3

 Add 10 to each side Now we need to clear the fraction Multiply each side by the common denominator (4) x 120

Check by substituting into the original inequality