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

	We have a fraction. To clear it, multiply by the common denominator which is 13
	Multiply each side by the common denominator
x > -78

Check by substituting into the original inequality

Example 2

	We have a fraction. To clear it, multiply by the common denominator which is -4
	Multiply each side by the common denominator (remember, to reverse the inequality symbol since we're multiplying by a negative)
-40 <y	Notice that this can be read "y is greater than -40"

Check by substituting into the original inequality

Example 3

	We have a fraction. To clear it, multiply by the common denominator which is 36
	Multiply each side by the common denominator
4 (-4) < -5r (3)	Reduce 36 and 9 on the left side to get 4 and reduce 12 and 36 on the right side to get 3. Then, do the multiplying.
-16 < -15r
	Divide each side by -15 (remember to reverse the symbol)

Check by substituting into the original inequality