We must find the inequality that represents the points in the shaded region of the plane. From the graph, we see that the points are strictly over a line. Let's find the equation of the dotted line.
We see that the line passes through the points:
• P1 = (x1,y1) = (5,4),
,
• P2 = (x2,y2) = (3,1).
The slope of the line is:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{1-4}{3-5}=\frac{-3}{-2}=\frac{3}{2}\text{.}[/tex]
The point-slope equation of the line is:
[tex]\begin{gathered} y=m\cdot(x-x_1)+y_1, \\ y=\frac{3}{2}\cdot(x-5)+4=\frac{3}{2}\cdot x-\frac{15}{2}+4=\frac{3}{2}\cdot x-\frac{7}{2}. \end{gathered}[/tex]
As we said above, the points are over the dotted line, so we have the following inequality:
[tex]\begin{gathered} y>\frac{3}{2}x-\frac{7}{2}, \\ 2y>3x-7, \\ 7>3x-2y. \end{gathered}[/tex]
Or:
[tex]3x-2y<7[/tex]
Answer
Last option:
[tex]3x-2y<7[/tex]
