Question

Graph each equation or inequality
$\text{-2x+5}\le 3y$

Short answer

The graph of the inequality $-2x+5\le 3y$ is

Step-by-step solution

Step-1 – Apply the concept of Linear Inequality

A linear inequality resembles a linear equation but with an inequality symbol instead of an equals symbol.

For example, $y\le x+5$is a linear inequality with $y=x+5$as a related linear equation.

Also, the graph of $y=x+5$separates the coordinate plane into two region. The line of the this graph acts as a boundary of the two separated region.

Step-2 – Calculate the boundary  

The given linear equality is $-2x+5\le 3y$ and thus the boundary which separates the coordinate plane into two region is role="math" localid="1646210171050" $-2x+5=3y$. As the inequality symbol is $\le$and therefore the boundary line will not be a dotted line which means that the line is also included.

Step-3 – Write in slope intercept form

The slope intercept form is in general given by $y=mx+b,\text{where}m=\frac{y_2-y_1}{x_2-x_1}$. Therefore, the slope intercept form of role="math" localid="1646210207734" $-2x+5=3y$ is obtained by keeping the variable y on one side and the rest (variable x, constants) on the other side.

Hence,

$\begin{array}{c}-2x+5=3y\\ 3y=-2x+5\text{(rewritingtheequation)}\\ \text{Dividethewholeequationby3,}\\ \frac{3y}{3}=\frac{1}{3}(-2x+5)\\ y=\frac{-2x}{3}+\frac{5}{3}\text{(dividingeachtermontherightsideby3)}\\ \end{array}$

The above equation is of the form $y=mx+b$and hence is of the slope intercept form.

Step-4 – Graph the boundary line

By using the slope intercept form graph the boundary line by finding the value of y corresponding to the value ofx.

Hence by using the values in the table, the graph is obtained as shown below:

Step-5 – Test a point on inequality

Choose a point which is not on a boundary line.

Since the point (0,0) is not on a boundary line, test the point $(0,0)$on the inequality $-2x+5\le 3y$.

Therefore,

$\begin{array}{c}-2\left(0\right)+5\le 3\left(0\right)\\ \Rightarrow 0+5\le 0\\ \Rightarrow 5\le 0\end{array}$

which is false.

Therefore, shade the region which does not contain $(0,0)$.

Hence the graph of the linear inequality is as shown below: