College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 1, Equations and Graphs - Section 1.7 - Solving Inequalities - 1.7 Exercises: 37

Answer

$[-3, 6]$ Refer to the image below for the graph.
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Work Step by Step

Factor the trinomial to obtain: $(x-6)(x+3)\le0$ Next, find the zeros of each factor. To find value/s of $x$ that will make each factor equal to zero, equate each factor to zero then solve each equation: $\begin{array}{ccc} &x-6=0 &\text{ or } &x+3=0 \\&x=6 &\text{ or } &x=-3 \end{array}$ Next step is to find the intervals. The zeros $-3$ and $6$ divide the number line into three intervals, namely: $(-\infty, -3), (-3, 6), \text{ and } (6, +\infty)$. $\bf\text{Make a table of signs}.$ (refer to the attached image below) $\bf\text{Solve}$ From the table of signs, it can be seen that $(x-6)(x+3)\le 0$, which is equivalent to $x^2-3x-18\le0$, in the interval $(-3, 6)$. The inequality involves $\le$ therefore the endpoints $-3$ and $6$ are part of the solution set. Thus, the solution set is $[-3, 6]$. To graph this, plot solid dots at $-3$ and $6$ then shade the region between them. (refer to the attached image in the answer part above)
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