which is the graph of linear inequality $2y > x - 2$?

Answer

Explanation:

Step1: Simplify the inequality

First, we simplify the given inequality (2y > x - 2) by dividing each term by 2. This gives us (y > \frac{1}{2}x - 1).

Step2: Analyze the boundary line

The boundary line for the inequality (y > \frac{1}{2}x - 1) is the line (y=\frac{1}{2}x - 1). Since the inequality is "greater than" (not "greater than or equal to"), the boundary line should be a dashed line. But looking at the options, all have a solid line? Wait, maybe in the problem's context, maybe it's a typo or maybe we proceed. The slope of the line (y = \frac{1}{2}x-1) is (\frac{1}{2}) and the y - intercept is (- 1).

Step3: Determine the shaded region

To find the shaded region, we can test a point. Let's test the point ((0,0)). Substitute (x = 0) and (y=0) into the inequality (y>\frac{1}{2}x - 1). We get (0>\frac{1}{2}(0)-1), which simplifies to (0 > - 1), which is true. So the point ((0,0)) should be in the shaded region.

Now let's analyze the options:

For the first graph: The shaded region is below the line? Let's check the y - intercept. The line (y=\frac{1}{2}x - 1) has y - intercept at ((0, - 1)). If we test ((0,0)) in the first graph's shaded region, is ((0,0)) shaded? In the first graph, the shaded region is below the line (since the line is going up and the shaded area is the lower part). But (0>\frac{1}{2}(0)-1) is true, but let's check the other options.
Second graph: The shaded region is above the line? Wait, no. Wait, let's re - express. The line (y=\frac{1}{2}x - 1): when (x = 0), (y=-1); when (y = 0), (x = 2). So the line passes through ((0,-1)) and ((2,0)). Now, for the inequality (y>\frac{1}{2}x - 1), the region above the line (y=\frac{1}{2}x - 1) should be shaded (because for a point above the line, (y) is greater than (\frac{1}{2}x-1)).

Looking at the options, the second graph (the middle one in the top row) has the shaded region above the line (y = \frac{1}{2}x-1) (since the line goes from ((0, - 1)) with slope (\frac{1}{2}), and the shaded area is above the line, and when we test ((0,0)), it is in the shaded area of the second graph. Also, the boundary line: since the inequality is (y>\frac{1}{2}x - 1), the line should be dashed, but if in the problem's options, the line is solid, maybe it's a mistake, but among the options, the second graph (the one in the middle of the top row) has the correct shaded region (above the line (y=\frac{1}{2}x - 1)) and the correct line (slope (\frac{1}{2}), y - intercept (-1)).