Q. 51 - 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

$51.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+2y\ge 1\\ x+2y\le 10\end{array}\right.$

Step-by-Step Solution

Answer

The shaded region shows the bounded region

step 1. introduction

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

$x+2y\ge 1$ ,the 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$(0,0)$ in the 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    $(0,0)$ does not contains the shaded portion in the equation  $x+2y\le 10$

step 4. The solution

The shaded portion is the bounded by all inequalities