Question

2-variable linear inequality
which linear inequality graphs the solution set shown?
a $y > \frac{3}{2}x + 3$
b $y \geq \frac{3}{2}x + 3$
c $y < \frac{2}{3}x + 3$
d $y \leq \frac{3}{2}x + 3$
which ordered pair is not a solution to the inequality?

Explanation:

Step1: Analyze the line's equation

The line in the graph has a y - intercept of 3 (since it crosses the y - axis at (0, 3)) and a slope. To find the slope, we can use two points on the line. Let's take (0, 3) and (-2, 0). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 3}{-2 - 0}=\frac{-3}{-2}=\frac{3}{2}\). So the equation of the line is \(y = \frac{3}{2}x+3\).

Step2: Analyze the line's style and the shaded region

The line is a solid line, which means the inequality includes equality (so the symbol is \(\geq\) or \(\leq\)). Now, we check the shaded region. We can test a point in the shaded region, for example, (0, 4). Plugging into the inequality \(y\) \(\_\_\) \(\frac{3}{2}x + 3\), when \(x = 0,y = 4\), and \(\frac{3}{2}(0)+3=3\). Since \(4>3\) and the line is solid, the inequality should be \(y\geq\frac{3}{2}x + 3\).

Answer:

B. \(y\geq\frac{3}{2}x + 3\)