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: 30

Answer

$(-\infty, 6]$ Refer to the graph below.
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Work Step by Step

Solve each inequality. \begin{align*} 7x+6&\le 48\\ 7x+6-6&\le 48-6\\ 7x&\le 42\\ \frac{7x}{7}&\le\frac{42}{7}\\ x&\le 6 \end{align*} \begin{align*} -4x&\ge -24\\ \frac{-4x}{-4} &\le \frac{-24}{-4}\\ x&\le 6 \end{align*} Thus, the given statement is equivalent to: $$x\le 6 \text{ and } x \le 6$$ 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. $x\le 6$ includes all the real numbers that are less than or equal $6$. The two sets are exactly the same, Hence, the solution set is $(-\infty, 6)$. Graph the solution set by plotting a solid dot at $4$ and then shading the region from $4$ to the left (Refer to the graph above.)
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