Precalculus (6th Edition) Blitzer

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

Chapter 7 - Review Exercises - Page 878: 40

Answer

The graph is shown below:
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Work Step by Step

Let us consider the inequality $ y\le -\frac{1}{2}x+2$; substitute the equal symbol in place of the inequality and rewrite the equation as given below: $ y=-\frac{1}{2}x+2$ To find the value of the x-intercept, put y = 0 as given below: $\begin{align} & 0=-\frac{1}{2}x+2 \\ & \frac{1}{2}x=2 \\ & x=4 \end{align}$ To find the value of the y-intercept, substitute x = 0 as given below: $\begin{align} & y=\left( -\frac{1}{2}\times 0 \right)+2 \\ & =2 \end{align}$ Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: $\begin{align} & y\le -\frac{1}{2}x+2 \\ & y+\frac{1}{2}x\le 2 \\ & 0+\left( \frac{1}{2}\times 0 \right)\le 2 \\ & 0\le 2 \end{align}$ So, the above expression is correct, which means the shaded region will contain the test point; that is, the region towards the origin will be shaded. Therefore, the line passes through points $\left( 4,0 \right)$ and $\left( 0,2 \right)$. Plot the graph using the intercepts as given below: Thus, the graph of the provided inequality is plotted and the line passes through points $\left( 4,0 \right)$ and $\left( 0,2 \right)$. Also, we see that the shaded region contains the origin.
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