Q. 49 - 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

$49.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ 2x+y\le 10\end{array}\right.$

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 $(0,0)$

$0+0\ge 2$

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 $(0,0)$

$$\begin{gathered} 0+0\le 8 \\ 0\le 8 \end{gathered}$$

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

Step 4. Consideration of fourth inequality

$2x+y\le 10$, the equation of line is by replacing inequality with $2x+y=10$

By using testing point $(0,0)$ in inequality$2x+y\le 10$

$$\begin{gathered} 0+0\le 10 \\ 0\le 10 \end{gathered}$$

The shaded part includes the test point $(0,0)$ in the inequality $2x+y\le 10$

 

Step 5. Solution

The shaded portion is the bounded by all inequalities