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: 47

Answer

The graph is shown below:
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Work Step by Step

Let us consider the inequalities $\begin{align} & 2x-y\ge 4 \\ & x+2y<2 \\ \end{align}$ Substitute the equals symbol in place of the inequality and rewrite the equation as given below: $\begin{align} & 2x-y=4 \\ & x+2y=2 \\ \end{align}$ To find the value of the x-intercept, substitute y = 0 as given below: $\begin{align} & 2x-y=4 \\ & 2x-0=4 \\ & x=2 \end{align}$ Therefore, the coordinates of the line are $\left( 2,0 \right)$. To find the value of the y-intercept, substitute x = 0 as given below: $\begin{align} & 2x-y=4 \\ & \left( 2\times 0 \right)-y=4 \\ & y=-4 \end{align}$ Therefore, the coordinates of the line are $\left( 0,-4 \right)$. Consider the second equation: $ x+2y=2$, To find the value of the x-intercept, substitute y = 0 as given below: $\begin{align} & x+2y=2 \\ & x+\left( 2\times 0 \right)=2 \\ & x=2 \end{align}$ Thus, the coordinates of the line are $\left( 2,0 \right)$. To find the value of the y-intercept, substitute x = 0 as given below: $\begin{align} & x+2y=2 \\ & 0+2y=2 \\ & y=1 \end{align}$ Therefore, the coordinates of the line are $\left( 0,1 \right)$. Then, take origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: $ 2x-y\ge 4\text{ and }x+2y<2$ $ 2*0-0\ge 4\text{ and }0+2*0<2$ $ 0\ge 4\text{ and }0<2$ So the origin is not included.
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