Q. 50 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

$50.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ x+2y\ge 1\end{array}\right.$

Step-by-Step Solution

Answer

The shaded region shows the bounded region

Step 1. Given information

Consider the first inequality in the system of inequalities is $x\ge 0$& $y\ge 0$

we get the shaded region

Step 2. Consideration of third inequality

$x+y\ge 2$, the equation of the equality consider by replacing with equality $x+y=2$

For determining the shaded portion we use to test a point with $(0,0)$

$$\begin{gathered} 0+0\ge 2 \\ 0\ge 2 \end{gathered}$$

The shaded part does not include the point $(0,0)$

Step 3. Consideration of fourth inequality

 $x+y\le 8$,the equation of line is replacing by inequality with equality$x+y=8$ 

By substituting by test point $(0,0)$

The point $(0,0)$ contains the shaded portion in the equation $x+y\le 8$

Step 4. Consideration of fourth inequality

$x+2y\ge 1$, the equation of line is by replacing inequality with $x+2y=1$

By using testing point$(0,0)$  in inequality $x+2y\ge 1$

$$\begin{gathered} 0+0\ge 1 \\ 0\ge 1 \end{gathered}$$

The shaded part does not includes the test point$(0,0)$  in the inequality  $x+2y\ge 1$

Step 5. Solution

The shaded portion is the bounded by all inequalities