Question

which system of inequalities is shown in the graph? show your work here \\( \bigcirc \\ y < 3 \text{ and } y > \frac{2}{3}x + 1 \\) \\( \bigcirc \\ y > 3 \text{ and } y > \frac{2}{3}x + 1 \\) \\( \bigcirc \\ y < 3 \text{ and } y < \frac{2}{3}x + 1 \\) \\( \bigcirc \\ y > 3 \text{ and } y < \frac{2}{3}x + 1 \\)

Explanation:

Step1: Analyze the horizontal line

The horizontal line in the graph is \( y = 3 \). The shaded region is below this line, so the inequality for this line is \( y < 3 \) (since the line is dashed or solid? Wait, looking at the graph, the horizontal line seems to be a boundary, and the shading is below, so \( y < 3 \).

Step2: Analyze the slanted line

The slanted line has a slope. Let's find its equation. From the graph, when \( x = 0 \), \( y = 1 \), and when \( x = 3 \), \( y = 3 \) (since \( \frac{2}{3}(3)+1 = 2 + 1 = 3 \)). So the equation is \( y=\frac{2}{3}x + 1 \). The shaded region is above this line (since the shading is between the horizontal line and above the slanted line). So the inequality for the slanted line is \( y>\frac{2}{3}x + 1 \).

Step3: Combine the inequalities

From Step1, \( y < 3 \), and from Step2, \( y>\frac{2}{3}x + 1 \). So the system is \( y < 3 \) and \( y>\frac{2}{3}x + 1 \).

Answer:

\( y < 3 \) and \( y > \frac{2}{3}x + 1 \) (the first option: \( y < 3 \) and \( y > \frac{2}{3}x + 1 \))