Question

this is the graph of 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 of the line

The line passes through \((0, -6)\) and \((7, 0)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\)? Wait, no, wait, let's check again. Wait, the y-intercept is at \((0, -6)\), and another point is \((7, 0)\)? Wait, no, when \(x = 7\), \(y = 0\), and when \(x = 0\), \(y=-6\). So the slope \(m=\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\)? Wait, no, wait, maybe I made a mistake. Wait, let's take two points: \((0, -6)\) and \((7, 0)\). So the change in \(y\) is \(0 - (-6)=6\), change in \(x\) is \(7 - 0 = 7\), so slope \(m=\frac{6}{7}\)? Wait, but let's check the line. Wait, the line goes from \((0, -6)\) to \((7, 0)\), so the slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\)? Wait, no, wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, let's re-express. Wait, the slope-intercept form is \(y = mx + b\), where \(b\) is the y-intercept. The y-intercept is \(-6\) (since the line crosses the y-axis at \((0, -6)\)). Now, let's find the slope. Let's take two points: \((0, -6)\) and \((7, 0)\). So the slope \(m=\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\)? Wait, no, wait, maybe I messed up the points. Wait, when \(x = 7\), \(y = 0\), and when \(x = 0\), \(y = -6\). So the slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\). Wait, but let's check the direction of the inequality. The shaded region: looking at the graph, the line is solid or dashed? Wait, the line is solid? Wait, no, the graph shows a line, and the shaded region? Wait, the graph has the line from \((0, -6)\) to \((7, 0)\), and the shaded area? Wait, the graph is a linear inequality. Let's see: the line is \(y=\frac{6}{7}x - 6\)? Wait, no, wait, maybe I made a mistake. Wait, let's take another point. Wait, when \(x = 7\), \(y = 0\), so plugging into \(y = mx - 6\), we get \(0 = 7m - 6\), so \(7m=6\), so \(m=\frac{6}{7}\). Wait, but the line is solid or dashed? The graph shows a solid line? Wait, the original graph: the line is solid? Wait, the shaded region: looking at the graph, the area above or below? Wait, the line goes from \((0, -6)\) to \((7, 0)\), and the shaded area? Wait, the graph's background is blue, and the line is a boundary. Wait, the inequality: let's check the slope again. Wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, maybe I made a mistake. Wait, let's re-express. The slope-intercept form is \(y = mx + b\). We know \(b = -6\) (y-intercept at \((0, -6)\)). Now, let's find the slope between \((0, -6)\) and \((7, 0)\): \(m=\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\). Now, the line is solid (since the graph shows a solid line? Wait, the original problem's graph: the line is solid? Wait, the user's graph: the line is a solid line? Wait, the shaded region: let's see, the area above or below? Wait, the graph has the line going from \((0, -6)\) to \((7, 0)\), and the shaded area? Wait, the blue area: looking at the graph, the region above the line? Wait, no, wait, the line is from \((0, -6)\) to \((7, 0)\), and the shaded area (the blue part) is above the line? Wait, no, let's check a test point. Let's take \((0, 0)\). Plug into the line equation \(y=\frac{6}{7}x - 6\). At \(x = 0\), \(y=-6\). So \((0, 0)\) is above the line (since \(0 > -6\)). Now, is the region above the line shaded? Wait, the graph's blue area: looking at the image, the blue area is above the line? Wait, no, the line is going from bottom left to top right, and the shaded area (the blue part) is above the line? Wait, no, the original graph: the line is a solid line, and the shaded region is above the line? Wait, but let's check the inequality. Wait, the line is \(y=\frac{6}{7}x - 6\)? Wait, no, wait, maybe I made a mistake in the slope. Wait, wait, let's take two points: \((0, -6)\) and \((7, 0)\). So the slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\), so the equation of the line is \(y=\frac{6}{7}x - 6\). Now, since the line is solid (not dashed), the inequality includes equality. Now, we need to determine the direction of the inequality. Let's take a test point, say \((0, 0)\). Plug into the inequality: \(0\) compared to \(\frac{6}{7}(0) - 6=-6\). So \(0 > -6\), so if the shaded region includes \((0, 0)\), then the inequality is \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, the graph's blue area: looking at the image, the blue area is above the line? Wait, the line is from \((0, -6)\) to \((7, 0)\), and the blue area is the region above the line? Wait, no, the original graph: the line is a solid line, and the shaded region (the blue part) is above the line? Wait, no, let's re-examine. Wait, the y-axis: the line crosses the y-axis at \((0, -6)\), and the x-axis at \((7, 0)\). The line is going up from left to right. The shaded area (the blue part) is above the line? Wait, no, the blue area is the background, and the line is a boundary. Wait, the problem says "this is the graph of a linear inequality". So first, find the equation of the line, then determine the inequality. The line is solid, so the inequality is either \(\geq\) or \(\leq\). Let's take a point in the shaded region. Let's take \((0, 0)\): plug into \(y\) and the line equation. The line at \(x = 0\) is \(y=-6\). So \(0\) is above \(-6\), so if the shaded region is above the line, then \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, the line equation is \(y=\frac{6}{7}x - 6\). Let's check the slope again. Wait, maybe I made a mistake in the slope. Wait, let's take two points: \((0, -6)\) and \((7, 0)\). The slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\), correct. So the line is \(y=\frac{6}{7}x - 6\). Now, the shaded region: let's see, the graph's blue area is above the line? Wait, the line is from \((0, -6)\) to \((7, 0)\), and the blue area is the region above the line? Wait, no, the blue area is the background, and the line is a boundary. Wait, the problem says "write the inequality in slope-intercept form". So first, the line is solid, so the inequality is either \(\geq\) or \(\leq\). Let's take a test point. Let's take \((0, 0)\). Plug into the line equation: \(y=\frac{6}{7}(0) - 6=-6\). So \((0, 0)\) is above the line (since \(0 > -6\)). Now, is \((0, 0)\) in the shaded region? Looking at the image, the blue area (the shaded region) includes \((0, 0)\), so the inequality is \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, maybe I messed up the slope. Wait, let's re-express. Wait, the line passes through \((0, -6)\) and \((7, 0)\), so the slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\), correct. So the line equation is \(y=\frac{6}{7}x - 6\). Now, the shaded region: since the line is solid, and the test point \((0, 0)\) satisfies \(y\geq\frac{6}{7}x - 6\) (because \(0\geq\frac{6}{7}(0)-6=-6\)), so the inequality is \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, let's check again. Wait, the line is from \((0, -6)\) to \((7, 0)\), so the slope is \(\frac{6}{7}\), y-intercept is \(-6\), so the line is \(y=\frac{6}{7}x - 6\). The shaded region is above the line, so the inequality is \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, maybe I made a mistake in the slope. Wait, wait, the line is actually \(y=\frac{6}{7}x - 6\)? Wait, no, wait, let's take another point. Let's take \(x = 7\), \(y = 0\): \(0=\frac{6}{7}(7)-6=6 - 6=0\), correct. So the line is correct. Now, the inequality: since the line is solid, and the shaded region is above the line, the inequality is \(y\geq\frac{6}{7}x - 6\)? Wait, no, wait, the graph's blue area: looking at the image, the blue area is above the line? Wait, no, the line is going from bottom left to top right, and the blue area (the shaded part) is above the line? Wait, no, the line is the boundary, and the shaded area is above it, so the inequality is \(y\geq\frac{6}{7}x - 6\). Wait, but let's check the slope again. Wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, maybe I made a mistake. Wait, the line is \(y=\frac{6}{7}x - 6\), and the inequality is \(y\geq\frac{6}{7}x - 6\). Wait, but let's confirm with the test point \((0, 0)\): \(0\geq\frac{6}{7}(0)-6=-6\), which is true. So that's the inequality.

Wait, no, wait, maybe the slope is \(\frac{6}{7}\)? Wait, no, wait, let's re-express. Wait, the line is \(y=\frac{6}{7}x - 6\), and the inequality is \(y\geq\frac{6}{7}x - 6\). Wait, but let's check the graph again. The line is solid, so the inequality includes equality. The shaded region is above the line, so the inequality is \(y\geq\frac{6}{7}x - 6\).

Wait, but maybe I made a mistake in the slope. Wait, let's take two points: \((0, -6)\) and \((7, 0)\). The slope is \(\frac{0 - (-6)}{7 - 0}=\frac{6}{7}\), correct. So the line is \(y=\frac{6}{7}x - 6\), and the inequality is \(y\geq\frac{6}{7}x - 6\).

Answer:

\(y \geq \frac{6}{7}x - 6\)