Given: A shaded portion of the graph
Required: System of inequalities to represnt the shaded portion.
Explanation:
Firstly consider the the non-dotted line.
It passes through two points. (0,3) and (-1.5,0)
So write the equation of line using two point form.
[tex]\begin{gathered} y-3=\frac{3-0}{0-(-1.5)}(x-0) \\ y-3=2x \end{gathered}[/tex]
so the equation is
[tex]2x-y+3=0[/tex]
Since (0,0) lies in shaded region, therefore the inequality is
[tex]2x-y+3\ge0[/tex]
Now, consider the dotted line.
It passes through two points. (-3,0) and (0,-1).
Equation of dotted line is
[tex]\begin{gathered} y-0=\frac{0-(-1)}{-3-0}(x+3) \\ y=-\frac{1}{3}(x+3) \end{gathered}[/tex]
So the equation is
[tex]\begin{gathered} 3y=-x-3 \\ x+3y+3=0 \end{gathered}[/tex]
Now, since (0,0) lies in the shaded portion, therefore the inequality is
[tex]x+3y+3>0[/tex]
Final Answer: The system of inequalities are
[tex]\begin{gathered} 2x-y+3\ge0 \\ x+3y+3>0 \end{gathered}[/tex]
