Precalculus (6th Edition) Blitzer

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

Chapter 7 - Test - Page 879: 8

Answer

The graph is shown below:

Work Step by Step

Let us consider the given inequalities $\begin{align} & 3x+y\le 9 \\ & 2x+3y\ge 6 \\ & x\ge 0 \\ & y\ge 0 \end{align}$ Put the equals symbol in place of the inequality and rewrite the equation as given below: By finding any two solutions of the linear equation, plot the graph of the linear equation $3x+y=9$: To find the value of the x-intercept, put the value of y = 0 as given below: $\begin{align} & 3x+0=9 \\ & x=\frac{9}{3} \\ & x=3 \end{align}$ To find the value of the y-intercept, put the value of x = 0 as given below: $\begin{align} & 3\left( 0 \right)+y=9 \\ & y=9 \end{align}$ Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,9 \right)$ and draw a solid line passing through these points. In the inequality there is the $\le $ symbol in which the equality is included. Now, this solid line divides the plane in three regions -- the line itself and two half-planes. Now, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: Check if the test point satisfies the inequality. $\begin{align} 3x+y\le 9 & \\ 3\left( 0 \right)+0\overset{?}{\mathop{\le }}\,9 & \\ 0\le 9 & \\ \end{align}$ Since the test point satisfies the inequality, shade the half-plane containing that test point towards the origin. Plot the graph using the intercepts as given below: By finding any two solutions of the linear equation, plot the graph of the linear equation $2x+3y\ge 6$: To find the value of the x-intercept, put the value of y = 0 as given below: $\begin{align} & 2x+3\left( 0 \right)=6 \\ & 2x=6 \\ & x=\frac{6}{2} \\ & x=3 \end{align}$ To find the value of the y-intercept, put the value of x = 0 as given below: $\begin{align} & 2\left( 0 \right)+3y=6 \\ & 3y=6 \\ & y=\frac{6}{3} \\ & y=2 \end{align}$ Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,2 \right)$ and draw a solid line passing through these points; since the inequality contains the $\le $ symbol, the equality is included. Now, this dashed line divides the plane in 3 region: the line itself, and the two half planes. Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: Check if the test point satisfies the inequality. $\begin{align} & 2x+3y\ge 6 \\ & 0+0\overset{?}{\mathop{>}}\,9 \\ & 0<9 \\ \end{align}$ Since, the test point does not satisfy the inequality, shade the half-plane not containing that test point that is away from the origin. And to plot the equation $ x\ge 0$, graph the line $ x=0$ and shade the right part of the line; that is, shade the region for which x values will be positive, as the inequality contains the $\ge $ symbol. Similarly, to plot the inequality $ y\ge 0$, graph the line $ y=0$ and shade the right part above the line; that is, shade the region for which y values will be positive, as the inequality contains the $\ge $ symbol. See the final graph below.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.