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: 46

Answer

The graph is shown below:
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Work Step by Step

Let us consider the inequalities $\begin{align} & 3x+2y\ge 6 \\ & 2x+y\ge 6 \end{align}$ Substitute the equals symbol in place of the inequality and rewrite the equation as given below: $\begin{align} & 3x+2y=6 \\ & 2x+y=6 \end{align}$ To find the value of the x-intercept, substitute y = 0 as given below: $\begin{align} & 3x+2y=6 \\ & 3x+2\times 0=6 \\ & 3x=6 \\ & x=2 \end{align}$ To find the value of the y-intercept, substitute x = 0 as given below: $\begin{align} & 3x+2y=6 \\ & 3\times 0+2y=6 \\ & 2y=6 \\ & y=3 \end{align}$ To find another value of the x-intercept, substitute y = 0 as given below: $\begin{align} & 2x+y=6 \\ & 2x+0=6 \\ & 2x=6 \\ & x=3 \end{align}$ To find another value of the y-intercept, substitute x = 0 as given below: $\begin{align} & 2x+y=6 \\ & \left( 2\times 0 \right)+y=6 \\ & y=6 \end{align}$ Then, take origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: $\begin{align} & 3x+2y\ge 6\text{ and }2x+y\ge 6 \\ & 0\ge 6\text{ and }0\ge 6 \\ \end{align}$ As we see that both above expressions are incorrect, we know that the shaded region will not contain the test point; that is, the region away from the origin must be shaded. Thus, the graph of the given inequality is plotted and the solution set lies in the shaded region.
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