Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.7 - Polynomial and Rational Inequalities - Exercise Set - Page 413: 66

Answer

$(-\infty,-7)\cup(1,\infty)$

Work Step by Step

Step 1. The given inequality can be separated into two cases: $x^2+6x+1\gt8$ and $x^2+6x+1\lt-8$ Step 2. For the first case, we have $x^2+6x-7\gt0$, $(x+7)(x-1)\gt0$ and the boundary points are $x=-7,1$ Using the test points to examine signs of the left side across the boundary points, we have $...(+)...(-7)...(-)...(1)...(+)...$ Thus the solutions are $x\lt-7$ or $x\gt1$, that is, $(-\infty,-7)\cup(1,\infty)$ Step 3. For the second case, we have $x^2+6x+9\lt0$, $(x+3)^2\lt0$ which has no solution. Step 4. Combining the above results, we have $(-\infty,-7)\cup(1,\infty)$ Step 5. The results are shown on a real number line in the figure.
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