Chapter 7.4, Problem 30E

Solution

The system of inequality, whose solution is the region shaded in pink in the given figure and all the boundaries are to be included.

The system of inequalities is $y\ge -2.5x+6,y\ge -\frac{1}{3}x+\frac{5}{3},x\ge 0,y\ge 0$ .

Given Information:

The figure is show as,

  

Explanation:

Consider the given system of equations,

  

The x -intercept is $\left(5,0\right)$ and the y -intercept is $\left(0,6\right)$ .

First find the slope between $\left(0,6\right)$ and $\left(2,1\right)$ .

  $\begin{array}{c}{m}_1=\frac{1-6}{2-0}\\ =-2.5\end{array}$

Now, find the line by using the points $\left(6,0\right)$ and $\left(2,1\right)$ .

  $\begin{array}{c}y-6=m_2\left(x-6\right)\\ y=-2.5\left(x-0\right)+6\\ y=-2.5x+6\end{array}$

Find the slopes between $\left(5,0\right)$ and $\left(2,1\right)$ .

  $\begin{array}{c}{m}_2=\frac{1-0}{2-5}\\ =-\frac{1}{3}\end{array}$

Now, find the line by using the points $\left(5,0\right)$ and $\left(2,1\right)$ .

  $\begin{array}{c}y-0=m_2\left(x-5\right)\\ y=-\frac{1}{3}\left(x-5\right)\\ y=-\frac{1}{3}x+\frac{5}{3}\end{array}$

It can be observed that the given shaded region is going to positive infinitely. So, the appropriate symbol for the equations is $\ge$ .

  $\begin{array}{c}y\ge -2.5x+6\\ y\ge -\frac{1}{3}x+\frac{5}{3}\\ x\ge 0\\ y\ge 0\end{array}$

Hence, the obtained system of inequalities are $y\ge -2.5x+6,y\ge -\frac{1}{3}x+\frac{5}{3},x\ge 0,y\ge 0$ .