Question

a linear inequality. write the inequality in slope-intercept form.
write your answer with y first, followed by an inequality symbol. use integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Find the slope-intercept form of the line

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

Step2: Determine the inequality symbol

The line is dashed, so the inequality is either \(<\) or \(>\). The shaded region is below the line (since the green area is below the dashed line), so we use the less than symbol. But we also need to check the direction. Wait, actually, when the line is dashed and the shading is below, the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, no, wait. Wait, let's check a test point. Let's take (0, 0), which is in the shaded region. Plug into the line equation: \(0=-\frac{2}{3}(0)+2\)? No, 0 < 2. Wait, but the line is \(y = -\frac{2}{3}x + 2\). If we plug (0,0) into \(y\) and the line: \(0\) vs \(-\frac{2}{3}(0)+2 = 2\). So 0 < 2, which is true. But wait, the shaded area is actually including points where \(y\) is less than the line? Wait, no, looking at the graph, the green area is below the dashed line. Wait, but let's check another point. Let's take (-3, 0). Plug into the line: \(y = -\frac{2}{3}(-3)+2 = 2 + 2 = 4\). The point (-3, 0): 0 < 4, which is true. Wait, but maybe I got the slope wrong. Wait, another way: the line goes from (0,2) to (3,0). So the slope is (0 - 2)/(3 - 0) = -2/3. Correct. Now, the dashed line means the inequality is strict (< or >). The shading is below the line, so for a linear inequality, if the line is \(y = mx + b\), and the shading is below, the inequality is \(y < mx + b\) (if the line is dashed) or \(y \leq mx + b\) (if solid). But in this case, the line is dashed, so it's \(y < -\frac{2}{3}x + 2\)? Wait, but wait, let's check the direction again. Wait, maybe I made a mistake. Wait, the y-intercept is 2, and the line goes down 2, right 3. So slope is -2/3. Now, the shaded region: let's take (0, -1), which is in the shaded area. Plug into the line: \(y = -\frac{2}{3}(0)+2 = 2\). So -1 < 2, which is true. So the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, but wait, maybe the other way. Wait, no, the standard form: if the line is dashed and the shading is below, it's \(y < mx + b\). But wait, let's check the graph again. The green area is to the left and below? Wait, no, the graph shows that for x negative, the y is higher? Wait, no, the line is decreasing. So as x increases, y decreases. So the shaded area is below the line, so the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, but maybe I messed up the direction. Wait, let's think of the slope-intercept form. The general form is \(y = mx + b\). If the line is dashed, and the shading is below, then \(y < mx + b\). But let's check the point (0,2) is on the line, but it's dashed, so not included. The shaded area includes points where y is less than the line. So the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, but wait, the problem says "Write the inequality in slope-intercept form" with y first. Wait, maybe I made a mistake in the inequality direction. Wait, let's look at the graph again. The green area is actually above or below? Wait, the y-axis: the green area is on the left side (negative x) and below? Wait, no, the green area is a large region. Wait, maybe the slope is actually positive? No, from (0,2) to (3,0), it's decreasing, so slope is negative. Wait, maybe the inequality is \(y \leq -\frac{2}{3}x + 2\)? But the line is dashed, so it should be strict. Wait, the problem says "dashed" line, so the inequality is < or >. Wait, maybe I got the shading wrong. Let's see: the green area is below the dashed line. So for a line \(y = mx + b\), if the shading is below, the inequality is \(y < mx + b\) (dashed) or \(y \leq mx + b\) (solid). Since it's dashed, it's \(y < -\frac{2}{3}x + 2\)? Wait, but let's check the point (0,0): 0 < 2, which is true. The point (3,0): 0 < -\frac{2}{3}(3)+2 = -2 + 2 = 0? No, 0 is not less than 0. Wait, that's a problem. Wait, (3,0) is on the line, which is dashed, so it's not included. So (3,0) is not in the shaded area. So when x=3, y=0 is on the line, not in the shade. So for x=3, y must be less than 0? But (3,0) is on the line, so y < 0? No, that can't be. Wait, I think I messed up the slope. Wait, maybe the line is from (0,2) to ( -3, 4)? Wait, no, the graph shows the line going from (0,2) to (3,0). Wait, maybe the slope is (0 - 2)/(3 - 0) = -2/3, correct. Wait, another approach: the equation of the line is \(y = -\frac{2}{3}x + 2\). The shaded region: let's take a point in the shaded region, say (-3, 2). Wait, (-3,2): plug into the line: \(y = -\frac{2}{3}(-3)+2 = 2 + 2 = 4\). So 2 < 4, which is true. So (-3,2) is in the shade. So y < -\frac{2}{3}x + 2. Wait, but when x=3, the line is y=0, so the shaded region at x=3 should have y < 0. But the graph shows the shaded region at x=3 is below y=0, which is correct. So the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, but the problem says "Write your answer with y first, followed by an inequality symbol". Wait, maybe I made a mistake in the direction. Wait, let's check the y-intercept. The line crosses the y-axis at (0,2), so b=2. Slope m=-2/3. So the line is \(y = -\frac{2}{3}x + 2\). The dashed line means the inequality is strict. The shading is below the line, so the inequality is \(y < -\frac{2}{3}x + 2\)? Wait, but let's check the point (0,0): 0 < 2, correct. (3, -1): -1 < -\frac{2}{3}(3)+2 = -2 + 2 = 0. So -1 < 0, correct. So yes, the inequality is \(y < -\frac{2}{3}x + 2\). Wait, but the problem's graph: the green area is actually including points where y is less than the line. So the answer should be \(y < -\frac{2}{3}x + 2\). Wait, but let me confirm again. The line is dashed, so no equality. Shading below, so y is less than the line. So the inequality is \(y < -\frac{2}{3}x + 2\).

Answer:

\(y < -\frac{2}{3}x + 2\)