Q. 47 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$

Step-by-Step Solution

Answer

The graph of $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$is bounded and the corner points are$(0,0), (0,2), \& (3,0)$  and the graph is 

Step 1. Given data

The given system of inequality is   

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$

Step 2. Properties of the graph of the inequality

Graph of $x=0 \& y=0$ will be y-axis and x-axis respectively

So region right side of the y-axis and above the x-axis will be shaded

Graph of $x+y=4$ will be a line

now test the point $(0,0)$ for inequality $x+y\le 4$

$$\begin{gathered} x+y\le 4 \\ 0=0\le 4 \\ 0<4 \end{gathered}$$

inequality satisfied so the origin will lie in the region of inequality

So the region toward origin from the line will be shaded

Step 3. Properties of the graph of the inequality

now test the origin point $(0,0)$ for inequality $2x+3y\le 6$

$$\begin{gathered} 2x+3y\le 6 \\ 2\left(0\right)+3\left(0\right)\le 6 \\ 0<6 \end{gathered}$$

inequality satisfied so the origin will lie in the region of inequality

So the region toward origin from the line will be shaded

Step 4. graph of the system of inequality

To plot the graph of $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$, Plot a solid line  $x+y=4$ and shade the region towards origin from the line, plot a solid line  $2x+3y=6$ and shade the region towards origin from the line and plot the y-axis and x-axis and shade the region right side of the y-axis and above the x-axis 

Graph of the system of inequalities state that the graph is bounded

Boundry lines of inequalities intersect at the points $(0,0), (0,2), \& (3,0)$so corner points are$(0,0), (0,2), \& (3,0)$