Question

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.y ≥ 3x - 6y < -2x - 1LABEL ALL COORDINATESAnswer

Answer 1

(-3, 2)

Explanation:

Given the system of inequalities:

`\begin{gathered} y\ge3x-6 \\ y<-2x-1 \end{gathered}`

To solve the inequalities graphically, follow the steps below:

Inequality 1

First, find the equation of the boundary line.

`y=3x-6`

Next, determine the intercepts to draw the boundary line.

`\begin{gathered} \text{When }x=0,y=3(0)-6=-6\implies(0,-6) \\ \text{When y}=0,0=3x-6\implies3x=6\implies x=2\implies(2,0) \end{gathered}`

Join the points (0,-6) and (2,0) using a solid line.

Finally, determine the required half-plane using the origin test.

`\begin{gathered} At\text{ (0,0)} \\ y\ge3x-6\implies0\ge-6(T\text{rue)} \end{gathered}`

The side that contains the point (0,0) is the required half-plane.

The graph showing the first inequality is given below:

Inequality 2

First, find the equation of the boundary line.

`y=-2x-1`

Next, determine the intercepts to draw the boundary line.

`\begin{gathered} \text{When }x=0,y=-2(0)-1=-1\implies(0,-1) \\ \text{When y}=0,0=-2x-1\implies-2x=1\implies x=-0.5\implies(-0.5,0) \end{gathered}`

Join the points (0,-1) and (-0.5,0) using a broken line.

Finally, determine the required half-plane using the origin test.

`\begin{gathered} At\text{ (0,0)} \\ y<2x-1\implies0<-1(False\text{)} \end{gathered}`

The side that DOES NOT contain the point (0,0) is the required half-plane.

The graph of the system of inequalities is given below:

A point in the solution set is (-3, 2).