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Simplifying Complex Fractions 1
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Solving Rational Inequalities with a Sign Graph
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Solving Rational Inequalities with a Sign Graph

Example

Solving a rational inequality

Solve and graph the solution set.

Solution

We do not multiply by the LCD as we do in solving equations. Instead, subtract from each side:

≥ 0  
≥ 0 Get a common denominator.
≥ 0 Simplify.
≥ 0  

Make a sign graph as shown in the figure below.

The computation of

involves multiplication and division. The result of this computation is positive if all of the three binomials are positive or if only one is positive and the other two are negative. The sign graph shows that this rational expression will have a positive value when x is between -4 and -1 and again when x is larger than 2. The solution set is (-4, -1) È [2, ). Note that -1 and -4 are not in the solution set because they make the denominator zero. The graph of the solution set is shown in the figure below.

 

Solving rational inequalities with a sign graph is summarized below.

 

Strategy for Solving a Rational Inequality with a Sign Graph

1. Rewrite the inequality with 0 on the right-hand side.

2. Use only addition and subtraction to get an equivalent inequality.

3. Factor the numerator and denominator if possible.

4. Make a sign graph showing where each factor is positive, negative, or zero.

5. Use the rules for multiplying and dividing signed numbers to determine the regions that satisfy the original inequality.

 
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