Question

ar equations and inequalities
which linear inequality is represented by the graph?
$y < \frac{2}{3}x + 3$
$y > \frac{3}{2}x + 3$
$y < \frac{3}{2}x + 3$
$y > \frac{2}{3}x + 3$

Explanation:

Step1: Find the slope of the line

The line passes through \((-4, 0)\) and \((0, 3)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 0}{0 - (-4)}=\frac{3}{4}\)? Wait, no, wait. Wait, looking at the graph, when \(x = 0\), \(y = 3\) (y-intercept \(b = 3\)). Let's take another point. From \((0,3)\) to \((3, 5)\)? Wait, no, the dashed line: let's check the slope. Wait, the line goes from \((-4, 0)\) to \((0, 3)\)? Wait, no, when \(x=-4\), \(y = 0\); when \(x = 0\), \(y = 3\). So slope \(m=\frac{3 - 0}{0 - (-4)}=\frac{3}{4}\)? No, that's not matching the options. Wait, the options have slopes \(\frac{2}{3}\) and \(\frac{3}{2}\). Wait, maybe I misread the points. Let's see, the line passes through \((0, 3)\) and \((3, 5)\)? Wait, when \(x = 3\), \(y = 5\)? No, the graph: let's count the rise over run. From \((0,3)\) to \((3, 5)\): rise is \(2\), run is \(3\)? No, that's \(\frac{2}{3}\). Wait, no, maybe \((0,3)\) to \((2, 5)\)? No, the options have \(\frac{2}{3}\) and \(\frac{3}{2}\). Wait, let's re - examine. The line: when \(x = 0\), \(y = 3\) (y - intercept \(b = 3\)). Let's take two points on the dashed line: \((-4, 0)\) and \((0, 3)\). Wait, the slope between \((-4,0)\) and \((0,3)\) is \(\frac{3 - 0}{0 - (-4)}=\frac{3}{4}\), which is not in the options. Wait, maybe I made a mistake. Wait, the options are \(y<\frac{2}{3}x + 3\), \(y>\frac{3}{2}x + 3\), \(y<\frac{3}{2}x + 3\), \(y>\frac{2}{3}x + 3\). Wait, let's check the slope again. Let's take the line: from \((0,3)\) to \((2, 5)\): rise \(2\), run \(2\)? No, that's \(1\). Wait, no, maybe the line is \(y=\frac{2}{3}x + 3\) or \(y=\frac{3}{2}x + 3\). Wait, let's check the direction of the shading. The shaded region is above the dashed line? Wait, the shaded area is above the line? Wait, the graph has the shaded region above the dashed line? Wait, the dashed line: if the line is dashed, it's a strict inequality. Now, let's find the slope correctly. Let's take two points on the line: \((0, 3)\) and \((3, 5)\): slope \(m=\frac{5 - 3}{3 - 0}=\frac{2}{3}\). Wait, no, \(5 - 3 = 2\), \(3 - 0 = 3\), so \(m=\frac{2}{3}\). Wait, but the options also have \(\frac{3}{2}\). Wait, maybe another pair of points: \((0,3)\) and \((2, 6)\)? No, that's \(m=\frac{3}{2}\). Wait, the line in the graph: when \(x = 2\), what's \(y\)? Let's see the grid. The dashed line: at \(x = 0\), \(y = 3\); at \(x = 2\), \(y = 3+\frac{3}{2}\times2=6\)? No, that's not. Wait, maybe I messed up the shading. The shaded area is above the dashed line? Wait, the first option: \(y<\frac{2}{3}x + 3\) (shaded below), \(y>\frac{3}{2}x + 3\) (shaded above, slope \(\frac{3}{2}\)), \(y<\frac{3}{2}x + 3\) (shaded below, slope \(\frac{3}{2}\)), \(y>\frac{2}{3}x + 3\) (shaded above, slope \(\frac{2}{3}\)). Let's find the correct slope. Let's take two points on the dashed line: \((0, 3)\) and \((3, 5)\): slope \(m=\frac{5 - 3}{3 - 0}=\frac{2}{3}\). Wait, but the options have \(\frac{2}{3}\) and \(\frac{3}{2}\). Wait, maybe the line is \(y=\frac{2}{3}x + 3\)? No, wait, when \(x = 3\), \(y=\frac{2}{3}\times3+3 = 2 + 3=5\), which matches the point \((3,5)\) on the dashed line. Now, the shading: the shaded area is above the dashed line? Wait, the graph shows the shaded region above the dashed line (the upper part). So the inequality should be \(y>\) the line. Now, the slope: we found \(m=\frac{2}{3}\)? Wait, no, wait, \((0,3)\) and \((3,5)\): \(y\) increases by \(2\) when \(x\) increases by \(3\), so slope \(\frac{2}{3}\). But the last option is \(y>\frac{2}{3}x + 3\). Wait, but let's check the slope again. Wait, maybe I made a mistake in the points. Let's take \((-4, 0)\) and \((0, 3)\): slope is \(\frac{3-0}{0 - (-4)}=\frac{3}{4}\), which is not matching. Wait, the options are:

1. \(y<\frac{2}{3}x + 3\)
2. \(y>\frac{3}{2}x + 3\)
3. \(y<\frac{3}{2}x + 3\)
4. \(y>\frac{2}{3}x + 3\)

Wait, let's calculate the slope correctly. Let's use the two points \((0, 3)\) and \((3, 5)\): slope \(m=\frac{5 - 3}{3 - 0}=\frac{2}{3}\). Now, the line is dashed, so the inequality is either \(y>\) or \(y<\). The shaded region: looking at the graph, the shaded area is above the dashed line (because when \(x = 0\), the shaded area is above \(y = 3\)? Wait, no, the graph's shaded region: the left - hand side, when \(x\) is negative, the shaded area is above the line? Wait, no, let's take a test point. Let's take \((0, 4)\), which is in the shaded region. Plug into the options:

- For \(y<\frac{2}{3}x + 3\): \(4<\frac{2}{3}(0)+3\)? \(4 < 3\)? No.
- For \(y>\frac{3}{2}x + 3\): \(4>\frac{3}{2}(0)+3\)? \(4>3\)? Yes, but slope is \(\frac{3}{2}\), which we didn't get.
- For \(y<\frac{3}{2}x + 3\): \(4<\frac{3}{2}(0)+3\)? \(4 < 3\)? No.
- For \(y>\frac{2}{3}x + 3\): \(4>\frac{2}{3}(0)+3\)? \(4>3\)? Yes. And the slope is \(\frac{2}{3}\), which matches the line \(y=\frac{2}{3}x + 3\) (since when \(x = 3\), \(y=\frac{2}{3}\times3+3 = 5\), which is on the dashed line). Wait, but earlier when I thought the slope was \(\frac{3}{2}\), that was a mistake. So the correct slope is \(\frac{2}{3}\), and the shading is above the line (so \(y>\)), so the inequality is \(y>\frac{2}{3}x + 3\).

Wait, but let's re - check the slope. Let's take two points on the dashed line: \((-4, 0)\) and \((0, 3)\). Slope \(m=\frac{3 - 0}{0 - (-4)}=\frac{3}{4}\), which is not matching. Wait, I must have misread the points. Wait, the dashed line: when \(x=-3\), \(y = 1\)? No, the y - intercept is \(3\) (at \(x = 0\), \(y = 3\)). Let's look at the grid again. The dashed line passes through \((0, 3)\) and \((3, 5)\) (since from \((0,3)\) moving 3 units right (x from 0 to 3) and 2 units up (y from 3 to 5), so slope \(\frac{2}{3}\)). Then the shaded region is above the line, so the inequality is \(y>\frac{2}{3}x + 3\), which is the last option.

Step2: Determine the inequality sign

The line is dashed, so the inequality is strict (\(>\) or \(<\)). The shaded region is above the line, so we use \(y>\) the line equation. The line equation is \(y=\frac{2}{3}x + 3\) (since slope \(m=\frac{2}{3}\) and \(b = 3\)). So the inequality is \(y>\frac{2}{3}x + 3\).

Answer:

\(y>\frac{2}{3}x + 3\) (the fourth option: \(y>\frac{2}{3}x + 3\))