Question

which inequality matches the graph? (1 point)
options:
-2x + 3y > 7
2x + 3y < 7
-3x + 2y > 7
3x - 2y < 7

Explanation:

Step1: Find the slope and intercept

The dashed line in the graph has a slope. Let's find two points on the line. From the graph, we can see that when \(x = 0\), \(y=-3.5\) (or \(-\frac{7}{2}\)) and when \(y = 0\), \(x=-\frac{7}{2}\)? Wait, no, let's re - examine. Wait, let's rewrite the inequalities in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).

For option A: \(-2x + 3y>7\), solve for \(y\):
\(3y>2x + 7\)
\(y>\frac{2}{3}x+\frac{7}{3}\approx\frac{2}{3}x + 2.33\)

For option B: \(2x + 3y<7\), solve for \(y\):
\(3y<-2x + 7\)
\(y<-\frac{2}{3}x+\frac{7}{3}\approx-\frac{2}{3}x + 2.33\)

For option C: \(-3x + 2y>7\), solve for \(y\):
\(2y>3x + 7\)
\(y>\frac{3}{2}x+\frac{7}{2}=1.5x + 3.5\)

For option D: \(3x-2y<7\), solve for \(y\):
\(-2y<-3x + 7\)
\(y>\frac{3}{2}x-\frac{7}{2}=1.5x-3.5\)

Now, looking at the graph, the y - intercept of the dashed line is \(-3.5\) (i.e., \(-\frac{7}{2}\)) and the slope is \(\frac{3}{2}\) (since for a line \(y = mx + b\), if we move 2 units in the x - direction, we move 3 units in the y - direction). Also, the shaded region is above the dashed line (because the line is dashed, so the inequality is strict, and the shading is above, so the inequality sign should be \(y>\cdots\)).

Step2: Analyze each option

• Option A: The slope is \(\frac{2}{3}\) and y - intercept is \(\frac{7}{3}\approx2.33\), which does not match the y - intercept of \(-3.5\) from the graph.
• Option B: The slope is \(-\frac{2}{3}\) and y - intercept is \(\frac{7}{3}\approx2.33\), the slope is negative, which does not match the positive slope of the line in the graph.
• Option C: The slope is \(\frac{3}{2}\) and y - intercept is \(\frac{7}{2} = 3.5\), the y - intercept is positive \(3.5\), but in the graph, the y - intercept is negative \(-3.5\).
• Option D: The slope is \(\frac{3}{2}\) and y - intercept is \(-\frac{7}{2}=-3.5\), and the inequality is \(y>\frac{3}{2}x-\frac{7}{2}\), which means the region above the line (since the line is dashed and we have \(y>\cdots\)) is shaded, which matches the graph.

Answer:

D. \(3x - 2y<7\)