Chapter 3.3, Problem 56PPSE

Solution

To find: To write the system of inequality for the graph given.

  $\begin{array}{l}y\le 3x+9\\ y\le 3\\ y\ge -3x+9\\ y\le 0\end{array}$

Given information: To write an equality for the graph given below:

  

Calculation:

From the graph, find the slope of $\left(-3,0\right)$ and $\left(-2,3\right)$ :

Use slope formula to find the slope $\left(m\right)$ using the slope formula:

  $m=\frac{3-0}{-2-\left(-3\right)}=3$

Now use point-slope form of line, $y-y_1=m\left(x-x_1\right)$ , to find the equation of the boundary line joining $\left(-3,0\right)$ and $\left(-2,3\right)$ :

  $\begin{array}{l}y-0=3\left(x-\left(-3\right)\right)\\ \Rightarrow y=3x+9\end{array}$

The boundary line is the graph of $y=3x+9$

Since the solution is shaded below the boundary line and the boundary line is solid, the inequality will be:

  $y\le 3x+9$

From the graph, find the equation of the boundary line joining the points $\left(-2,3\right)$ and $\left(2,3\right)$ will be:

  $y=3$

Since the solution is shaded below the boundary line and the boundary line is solid, the inequality will be:

  $y\le 3$

From the graph, find the slope of $\left(2,3\right)$ and $\left(3,0\right)$ :

Use slope formula to find the slope $\left(m\right)$ using the slope formula:

  $m=\frac{0-3}{3-2}=-3$

Now use point-slope form of line, $y-y_1=m\left(x-x_1\right)$ , to find the equation of the boundary line joining $\left(2,3\right)$ and $\left(3,0\right)$ :

  $\begin{array}{l}y-0=-3\left(x-2\right)\\ \Rightarrow y=-3x+9\end{array}$

The boundary line is the graph of $y=-3x+9$

Since the solution is shaded above the boundary line and the boundary line is solid, the inequality will be:

  $y\ge -3x+9$

From the graph, find the equation of the boundary line joining the points $\left(-3,0\right)$ and $\left(3,0\right)$ will be:

  $y=0$

Since the solution is shaded below the boundary line and the boundary line is solid, the inequality will be:

  $y\le 0$