Q. 52 - Precalculus Enhanced with Graphing Utilities

Question

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

 $52.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+2y\ge 1\\ x+2y\le 10\\ x+y\ge 2\\ x+y\le 8\end{array}\right.$

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 $x\ge 0$ & $y\ge 0$.

Step 2. Consideration of third inequality

$x+2y\ge 1$ equation of the equality consider by replacing with equality $x+2y=1$

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

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

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

Step 3. Consideration of fourth inequality

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

By substituting by test point $(0,0)$

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

The point $(0,0)$ contains the shaded portion in the equation  $x+2y\le 10$

Step 4. Consideration of five inequality

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

By using the testing point $(0,0)$

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

The point $(0,0)$ does not includes with the inequality $x+y\ge 2$

Step 5. Consideration of sixth inequality

$x+y\le 8$,the equation of line replace with equality $x+y=8$

By using test point$(0,0)$

$0+0\le 8$

$0\le 8$

The point $(0,0)$includes the inequality $x+y\le 8$ 

Step 6. Solution

The shaded portion is the bounded region: