Q7. - Algebra 2

Question

Graph each inequality. 

 

$x-2y\le 5$

Step-by-Step Solution

Answer

The graph of the inequality $x-2y\le 5$ is provided below.

Step-1 – Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid. 

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality. 

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-2 – Interpret the sign of the inequality

Consider the inequality provided below.

$x-2y\le 5$

The inequality contains the sign of less than or equal to.

Therefore, the boundary line will be solid.

Step-3 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-4 –Write the equations in slope-intercept form

Consider the equation $x-2y=5$. 

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $x$ and then divide by $-2$.

$\begin{array}{c}x-2y-x=5-x\\ -2y=-x+5\\ y=\frac{1}{2}x-\frac{5}{2}\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{1}{2}$ and intercept of y-axis c is $-\frac{5}{2}$.

Step-5 – Graph the inequality

The corresponding equation is $x-2y=5$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

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

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Thus, the shaded region is provided below.

Thus, the graph of the inequality is shown above.