Chapter 3.3, Problem 25PPSE

Solution

To solve: The system of inequality using the graphing method $\left(\begin{array}{l}y\le -\frac{3}{4}x\\ y\ge -\left|x\right|\end{array}\right.$ .

Given information:

The given system of inequalities is $\left(\begin{array}{l}y\le -\frac{3}{4}x\\ y\ge -\left|x\right|\end{array}\right.$ .

Concept used:

Graph each inequality and then overlapping region is the solution of the system.

Plot the graph:

The graph of the inequality $y\le -\frac{3}{4}x$ is given below.

  

The graph of the inequality $y\ge -\left|x\right|$ is given below:

  

The graph of the system of inequalities $\left(\begin{array}{l}y\le -\frac{3}{4}x\\ y\ge -\left|x\right|\end{array}\right.$ is given below:

  

Now to check any two point from the overlapping region and check the solution region is true or not.

Take the point $\left(2,-2\right)$

For $\left(2,-2\right)$ check $y\le -\frac{3}{4}x\text{and}\text{y}\ge -\left|x\right|$ :

  $-2\le \frac{-3}{2},-2\ge -2$ ,both inequalities are true to the point $\left(2,3\right)$ .

So the overlapping region is the solution of the system of given inequalities.