Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.8 - Inequalities - 1.8 Exercises - Page 89: 78


$\left( -\infty, -2 \right] \cup \left[ 2,\infty \right)$

Work Step by Step

Using the properties of inequality, the given expression, $ \dfrac{1}{2}|x|\ge1 ,$ is equivalent to \begin{array}{l}\require{cancel} 2\cdot\dfrac{1}{2}|x|\ge1\cdot2 \\\\ |x|\ge2 .\end{array} For any $a\gt0$, $|x|\gt a$ implies $x\gt a \text{ OR } x\lt -a$. (The symbol $\gt$ may be replaced with $\ge$.) Using the concept above, the solutions to the given inequality, $ |x|\ge2 ,$ is \begin{array}{l}\require{cancel} x\ge2 ,\\\\\text{ OR }\\\\ x\le-2 .\end{array} In interval notation, the solution set is $ \left( -\infty, -2 \right] \cup \left[ 2,\infty \right) .$
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