Q. 54 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 53–56, write a system of linear inequalities for the given graph.

54.

Step-by-Step Solution

Answer

The inequalities are

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x\le 6\\ y\le 5\\ x+y\ge 2\end{array}\right.$

Step 1. Given information

The graph is in first quadrant so the values are $x\ge 0$&$y\ge 0$

Step 2. About parallel lines

The lines parallel to y-axis has equation $x=k$

here $k=6$ then the equation is $x=6$

the lines parallel to x-axis has equation $y=k$

here k=5 then the equation is $y=5$

Due to origin not included in the region so the inequalities are $x\le 6$&$y\le 5$

Step 3. About line equation of given points

Given the points are $(0,2)$&$(2,0)$

slope:$m=(\raisebox{1ex}{$y_2-y_1$}\!\left/ \!\raisebox{-1ex}{$(x_2-x1)$}\right.$

$$\begin{gathered} m=\raisebox{1ex}{$(2-0)$}\!\left/ \!\raisebox{-1ex}{$(0-2)$}\right. \\ m=-1 \end{gathered}$$

The line equation is:

 $$\begin{gathered} y-y_1=m(x-x_1) \\ y-0=-1(x-2) \\ y=-x+2 \\ x+y=2 \end{gathered}$$

The line excludes the origin so the inequality $x+y\ge 2$.

Atep 4. Solution

The inequalities bounded by the given graph

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x\le 6\\ y\le 5\\ x+y\ge 2\end{array}\right.$