Answer:
(-3, 2)
Explanation:
Given the system of inequalities:
[tex]\begin{gathered} y\ge3x-6 \\ y<-2x-1 \end{gathered}[/tex]To solve the inequalities graphically, follow the steps below:
Inequality 1
First, find the equation of the boundary line.
[tex]y=3x-6[/tex]Next, determine the intercepts to draw the boundary line.
[tex]\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}[/tex]Join the points (0,-6) and (2,0) using a solid line.
Finally, determine the required half-plane using the origin test.
[tex]\begin{gathered} At\text{ (0,0)} \\ y\ge3x-6\implies0\ge-6(T\text{rue)} \end{gathered}[/tex]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.
[tex]y=-2x-1[/tex]Next, determine the intercepts to draw the boundary line.
[tex]\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}[/tex]Join the points (0,-1) and (-0.5,0) using a broken line.
Finally, determine the required half-plane using the origin test.
[tex]\begin{gathered} At\text{ (0,0)} \\ y<2x-1\implies0<-1(False\text{)} \end{gathered}[/tex]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).
