Question

this graph shows the solutions to the inequalities $y > \frac{3}{2}x - 2$ and $y < \frac{3}{2}x - 10$. does the system of inequalities have solutions? if so, which region contains the solutions?
graph with regions a, b, c
text description for graph
\bigcirc a. there is a solution, and it is shown by region c.
\bigcirc b. there is no solution.
\bigcirc c. there is a solution, and it is shown by region a.

Explanation:

Step1: Analyze the slopes of the lines

The two inequalities are \( y > \frac{3}{2}x - 2 \) and \( y < \frac{3}{2}x - 10 \). Both lines have the same slope \( m=\frac{3}{2} \), so they are parallel.

Step2: Analyze the regions of the inequalities

For \( y > \frac{3}{2}x - 2 \), the solution region is above the line \( y = \frac{3}{2}x - 2 \). For \( y < \frac{3}{2}x - 10 \), the solution region is below the line \( y=\frac{3}{2}x - 10 \). Since the line \( y = \frac{3}{2}x - 2 \) is above the line \( y=\frac{3}{2}x - 10 \) (because the y - intercept of \( y = \frac{3}{2}x - 2 \) is \(- 2\) and the y - intercept of \( y=\frac{3}{2}x - 10\) is \(-10\), and \(-2>-10\)), the region above \( y = \frac{3}{2}x - 2 \) and the region below \( y=\frac{3}{2}x - 10 \) do not overlap.

Answer:

B. There is no solution.