Chapter 6, Problem 1RE

In Exercises 1-4, sketch the region corresponding to the given inequalities, say whether it is bounded, and give the coordinates of all corner points.

$2x-3y\le 12$

To determine

To graph: The region that corresponds to the provided inequality $2x-3y\le 12$ and find the coordinates of all the corner points.

Explanation of Solution

Given Information:

The provided inequality is $2x-3y\le 12$.

Graph:

Consider the provided inequality:

$2x-3y\le 12$

Now, firstly consider the equation:

$2x-3y\le 12$

Now, substitute values of $x$ to obtain the corresponding values of $y$ and sketch the graph of the line.

For $x=0$;

$\begin{array}{c}2\left(0\right)-3y=12\\ -3y=12\\ y=-\frac{12}{3}\\ y=-4\end{array}$

Therefore, the point is $\left(0,-4\right)$.

For $y=0$:

$\begin{array}{c}2x-3\left(0\right)=12\\ 2x=12\\ x=\frac{12}{2}\\ x=6\end{array}$

Thus, the point is $\left(6,0\right)$.

Thus, the table for the corresponding $x$ and $y$ values for the provided equation of line is:

$x$	$6$	$0$
$y$	$0$	$-4$

Join these points on a graph to form a line as shown below:

Now, the equation of boundary line is $2x-3y\le 12$. It has $x$-intercept $\left(6,0\right)$ and $y$-intercept $\left(0,-4\right)$.

Now, use $\left(0,0\right)$ as test point because it is not on the line.

Consider the provided inequality:

$2x-3y\le 12$

At start, trace the sketch of line $2x-3y=12$

Test this sketched curve for $\left(0,0\right)$ in the provided inequality as:

$\begin{array}{l}2\left(0\right)-3\left(0\right)\le 12\\ 0\ne 12\end{array}$

So, the graph of the provided inequality is not satisfied.

Hence, the graph of inequality is not containing the test point and it is unbounded.

Interpretation:

The graph of inequality is not containing the test point and it is unbounded.