Question

which system of inequalities corresponds to the following graph: graph options: 1. $y > 2x + 1$; $y > \frac{1}{2}x - 2$ 2. $y < 2x + 1$; $y > \frac{1}{2}x - 2$ 3. $y > 2x + 1$; $y < \frac{1}{2}x - 2$

Explanation:

Step1: Analyze the first line (red dashed)

The red dashed line has a slope of 2 and y-intercept 1, so its equation is \( y = 2x + 1 \). The shaded region is below this line (since the blue area is below the red dashed line), so the inequality is \( y < 2x + 1 \).

Step2: Analyze the second line (blue solid)

The blue solid line has a slope of \( \frac{1}{2} \) and y-intercept -2, so its equation is \( y = \frac{1}{2}x - 2 \). The shaded region is above this line (since the blue area is above the blue solid line), so the inequality is \( y > \frac{1}{2}x - 2 \).

Step3: Match with options

The system of inequalities \( y < 2x + 1 \) and \( y > \frac{1}{2}x - 2 \) matches the second option.

Answer:

\( y < 2x + 1 \)
\( y > \frac{1}{2}x - 2 \) (the second option in the list)