College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.6 - Page 420: 70

Answer

$(-∞, -1) ∪ (1, 2) ∪ (3, ∞)$

Work Step by Step

Consider the Inequality as follows: $\frac{x^{2} -3x +2}{x^{2} -2x -3}$$> 0$ Here are the steps required for Solving Rational Inequalities: 1. One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. $x^{2} -3x +2 = 0$ This implies $ x =1,2$ and $x^{2} -2x -3 = 0$ This implies $x = -1, 3$ These solutions are used as boundary points on a number line. 2. Locate these boundary points on a number line found in Step 1 to divide the number line into intervals. The boundary points divide the number line into five intervals: $(-∞, -1), (-1, 1), (1, 2), (2, 3), (3, ∞)$ 3. Now, one test value within each interval is chosen and $f$ is evaluated at that number. Intervals: $(-∞, -1) (-1, 1) (1, 2) (2, 3) (3, ∞)$ Test value: $-2$ $0$ $1.5$ $ 2.5$ $4$ Sign Change: Positive Negative Positive Negative Positive $f (x) > 0?$: T F T F T 4. Write the solution set, selecting the interval or intervals that satisfy the given inequality. Based on our work done in Step 3, we see that $f (x) > 0$ for all $x$ in $(-∞, -1)$or $(1, 2)$ or$(3, ∞)$. Thus, the interval notation of the given inequality is $(-∞, -1) ∪ (1, 2) ∪ (3, ∞)$ and the graph of the solution set on a number line is shown as follows:
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