Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 1 - Section 1.6 - Set Operations and Compound Inequalities - 1.6 Exercises - Page 109: 26


$[2, 6]$ Refer to the graph below.

Work Step by Step

Solve each inequality. \begin{align*} x+5&\le 11\\ x+5-5&\le11-5\\ x&\le 6 \end{align*} \begin{align*} x-3&\ge -1\\ x-3+3&\ge-1+3\\ x&\ge 2 \end{align*} Thus, the given statement is equivalent to: $$x\le 6 \text{ and } x \ge 2$$ The conjunction "and" means intersection so find the intersection of the two sets. Recall: The intersection of sets $A$ and $B$ is the set that contains the elements that are common to both sets. $x\le6$ includes all the real numbers that are less than or equal to $6$. $x\ge2$ includes all the real numbers that are greater than or equal to $2$. Note the that numbers common to the two given sets are the numbers that are greater than or equal to $2$ but less than or equal to $6$. Hence, the solution set is $[2, 6]$. Graph the solution set by plotting solid dots at $2$ and $6$, then shading the region between them. (Refer to the graph above.)
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