Chapter 3.7, Problem 29WE

To determine

To draw: The graph of given inequality in a coordinate plane.

Answer to Problem 29WE

  

Explanation of Solution

Given information: The given inequality is:

  $\begin{array}{l}3x+2y\le 6(i)\\ 2x+3y\ge 6(ii)\end{array}$

Calculation: The given inequality is:

  $\begin{array}{l}3x+2y\le 6(i)\\ 2x+3y\ge 6(ii)\end{array}$ .

The corresponding equation is:

  $\begin{array}{l}3x+2y=6(iii)\\ 2x+3y=6(iv)\end{array}$

  $\begin{array}{l}3x+2y=6\\ \mathrm{put}x=0\\ 0+2y=6\\ y=\frac{6}{2}=3\\ \mathrm{put}y=0\\ 3x+2(0)=6\\ 3x-0=6\\ x=\frac{6}{3}=2\end{array}$

$x$	$0$	$2$
$y$	$3$	$0$

  $\begin{array}{l}2x+3y=6\\ \mathrm{put}x=0\\ 2(0)+3y=6\\ 3y=6\\ y=\frac{6}{3}=2\\ \mathrm{put}y=0\\ 2x+3(0)=6\\ 2x+0=6\\ 2x=\frac{6}{2}=3\end{array}$

$x$	$0$	$3$
$y$	$2$	$0$

  

Put $x=0\mathrm{and}y=0$ in equation $(i)$ we get:

  $\begin{array}{l}3(0)+2(0)\le 6\\ 0+0\le 6\\ 0\le 6\end{array}$

Which is true, therefore the closed plane containing the origin is graph of given inequality.

Put $x=0\mathrm{and}y=0$ in equation $(ii)$ we get:

  $\begin{array}{l}2(0)+3(0)\ge 6\\ 0+0\ge 6\\ 0\ge 6\end{array}$

Which is false, therefore the closed half plane does not contain the origin is graph of given inequality.

Thus the common shaded region is the graph of all inequality.