Question

Determine the system of inequalities that represents the shaded area .

Answer 1

For the upper line:

`\begin{gathered} (x1,y1)=(0,2) \\ (x2,y2)=(2,3) \\ m=(y2-y1)/(x2-x1)=(3-2)/(2-0)=(1)/(2) \\ \text{ using the point-slope equation:} \\ y-y1=m(x-x1) \\ y-2=(1)/(2)(x-0) \\ y=(1)/(2)x+2 \end{gathered}`

For the lower line:

`\begin{gathered} (x1,y1)=(0,-3) \\ (x2,y2)=(2,-2) \\ m=(-2-(-3))/(2)=(1)/(2) \\ \text{ Using the point-slope equation:} \\ y-y1=m(x-x1) \\ y-(-3)=(1)/(2)(x-0) \\ y+3=(1)/(2)x \\ y=(1)/(2)x-3 \end{gathered}`

Therefore, the system of inequalities is given by:

`\begin{gathered} y\le(1)/(2)x+2 \\ y\ge(1)/(2)x-3 \end{gathered}`