Question

Solve the system of two linear inequalities graphically. Graph the solution set of the first linear inequality. Choose the type of boundary line? Enter two points on the boundary line? Select a reason you wish to be shaded?

Answer 1

First line

We will graph the solution set of the first linear inequality.

`3y<7x-21`

The related equation is

`3y=7x-21`

Now we take any two values of x, plug them into the equation, and solve for their respective y-coordinates.

`\begin{gathered} \text{If }x=0 \\ 3y=7x-21 \\ 3y=7(0)-21 \\ 3y=-21 \\ \text{ Divide by 3 from both sides of the equation} \\ (3y)/(3)=-(21)/(3) \\ y=-7 \\ \text{ Then, the line passes through the point (0,-7)} \end{gathered}`

`\begin{gathered} \text{If }x=3 \\ 3y=7x-21 \\ 3y=7(3)-21 \\ 3y=21-21 \\ 3y=0 \\ \text{ Divide by 0 from both sides of the equation} \\ (3y)/(3)=(0)/(3) \\ y=0 \\ \text{ Then, the line passes through the point (3,0)} \end{gathered}`

Since the inequality is <, a strict one, the borderline is dashed.

Now, we consider a point that is not on the line, for example (0,0), and substitute in the inequality:

`\begin{gathered} 3y<7x-21 \\ 3(0)<7(0)-21 \\ 0<-21 \end{gathered}`

This is false. So, the solution does not contain the point (0,0). Therefore, we shade the lower half of the line.

Second line

We will graph the solution set of the second linear inequality.