Q18. - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

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 $-2x+5=3y$. 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 $-2x+5=3y$ is obtained by keeping the variable y on one side and the rest (variable x, constants) on the other side. 

Hence,

$\begin{array}{c}-2x+5=3y\\ 3y=-2x+5\text{ (rewriting the equation)}\\ \text{Divide the whole equation by 3,}\\ \frac{3y}{3}=\frac{1}{3}(-2x+5)\\ y=\frac{-2x}{3}+\frac{5}{3}\text{ (dividing each term on the right side by 3)}\\ \end{array}$

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 of x.

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: