Select a true statement to describe the graphed system of inequalities

Question

y > -1/3x and y < -1/3x – 2

A. The system has one solution
B. The system has infinite solutions
C. The system has two solutions
D. The system has no solutions

Answer 1

To analyze the system of inequalities given by:

1. \( y > -\frac{1}{3}x \)
2. \( y < -\frac{1}{3}x - 2 \)

we can break it down as follows:

1. The first inequality, \( y > -\frac{1}{3}x \), represents a region above the line \( y = -\frac{1}{3}x \).
2. The second inequality, \( y < -\frac{1}{3}x - 2 \), represents a region below the line \( y = -\frac{1}{3}x - 2 \).

Now, consider the slopes of the two lines:

• Both lines have the same slope of \(-\frac{1}{3}\), meaning they are parallel.
• The second line is vertically translated downwards by 2 units compared to the first line, making it impossible for any \( y \) value to simultaneously satisfy both inequalities simultaneously.

Given that there are no points that can satisfy \( y \) being both greater than the first line and less than the second line, we conclude:

D. The system has no solutions.