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: 49

Answer

The graph is shown below:
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Work Step by Step

Let us consider the inequalities $\begin{align} & x+y\le 6 \\ & y\ge 2x-3 \\ \end{align}$ Substitute the equals symbol in place of the inequality and rewrite the equation as given below: $\begin{align} & x+y=6 \\ & y=2x-3 \\ \end{align}$ To find the value of the x-intercept for the equation $ x+y=6$, substitute y = 0 as given below: $\begin{align} & x+y=6 \\ & x+0=6 \\ & x=6 \\ \end{align}$ Therefore, the coordinates of the line are $\left( 6,0 \right)$. To find the value of the y-intercept for equation $ x+y=6$, put the value of x = 0 as given below: $\begin{align} & x+y=6 \\ & 0+y=6 \\ & y=6 \\ \end{align}$ Thus, the coordinates of the line are $\left( 0,6 \right)$. To find the value of the x-intercept for the equation $ y=2x-3$, put the value of y = 0 as given below: $\begin{align} & y=2x-3 \\ & 0=2x-3 \\ & x=\frac{3}{2} \\ & x=1.5 \\ \end{align}$ Therefore, the coordinates of the line are $\left( 1.5,0 \right)$. To find the value of the y-intercept for the equation $ y=2x-3$, put the value of x = 0 as given below: $\begin{align} & y=2x-3 \\ & y=\left( 2\times 0 \right)-3 \\ & y=-3 \\ \end{align}$ Therefore, the coordinates of the line are $\left( 0,-3 \right)$. Then, take the origin (0,0) as a test point for the equations and check the region in the graph to shade: $\begin{align} & x+y\le 6\text{ and }y\ge 2x-3 \\ & 0+0\overset{?}{\mathop{\le }}\,6\text{ and 0}\overset{?}{\mathop{\ge }}\,\left( 2\times 0 \right)-3 \\ & 0\le 6\text{ and 0}\ge -3 \\ \end{align}$ Since, the test point satisfies both the inequalities, shade the region toward the test point for both inequalities. Thus, the graph of the provided inequality is plotted and the solution set lies in the shaded region.
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