Chapter 3, Problem 5STP

To determine

Find the solution region in the given graph for given inequalities.

Answer to Problem 5STP

Thesolution region is $\left(\text{III}\right)$ .

Explanation of Solution

Given:

The given inequalitiesare $y\le \frac{1}{2}x-2$ and $y\le -\frac{2}{3}x-1$ . The given graph is

  

Calculation:

Draw a graph of the equation $y=\frac{1}{2}x-2$ .

Find x-intercept $\left(y=0\right)$ .

  $\begin{array}{l}0=\frac{1}{2}x-2\\ \Rightarrow x=4\end{array}$

x-intercept is $\left(4,0\right)$ .

Find y-intercept $\left(x=0\right)$ .

  $\begin{array}{l}y=\frac{1}{2}\cdot 0-2\\ \Rightarrow y=-2\end{array}$

y-intercept is $\left(0,-2\right)$ .

Using these two points draw the graph of the equation $y=\frac{1}{2}x-2$ .

Check that the line is the part of the solution or not for the given inequality [test point $\left(0.857,-1.571\right)$ ].

  $\begin{array}{l}-1.571\le \frac{1}{2}\cdot 0.857-2\\ -1.571\le --1.571\end{array}$

Draw a graph of the equation $y=-\frac{2}{3}x-1$ .

Find x-intercept $\left(y=0\right)$ .

  $\begin{array}{l}0=-\frac{2}{3}x-1\\ \Rightarrow x=-\frac{3}{2}\end{array}$

x-intercept is $\left(-\frac{3}{2},0\right)$ .

Find y-intercept $\left(x=0\right)$ .

  $\begin{array}{l}y=-\frac{2}{3}\cdot 0-1\\ \Rightarrow y=-1\end{array}$

y-intercept is $\left(0,-1\right)$ .

Using these two points draw the graph of the equation $y=-\frac{2}{3}x-1$ .

Check that the origin is the part of the solution or not for the given inequality [test point $\left(0.857,-1.571\right)$ ].

  $\begin{array}{l}-1.571\le -\frac{2}{3}\cdot 0.857-1\\ -1.571\le -1.571\end{array}$

Therefore, lines are the part of the solution.

Hence the graph of the inequalities is given below.

  

From the graph the solution region is $\left(\text{III}\right)$ .