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

The line passes through points \((-8, 2)\) and \((8, -4)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-4 - 2}{8 - (-8)}=\frac{-6}{16}=-\frac{3}{8}\).

Step2: Find the y - intercept

The line crosses the y - axis at \((0, -1)\), so \(b=-1\).

Step3: Determine the inequality symbol

The line is solid? No, wait, the graph is shaded above or below? Wait, the line is blue and the shading is above? Wait, no, the line passes through \((-8,2)\) and \((8, -4)\), and the y - intercept is \((0, -1)\). The line is a straight line, and the shading is above? Wait, no, let's check the slope - intercept form \(y = mx + b\). The line equation is \(y=-\frac{3}{8}x - 1\). Now, the shaded region: let's take a test point, say \((0,0)\). Plug into the inequality: \(0\)? compared to \(-\frac{3}{8}(0)-1=-1\). Since \(0 > - 1\) and the line is dashed? Wait, no, the line in the graph: looking at the graph, the line is a solid line? Wait, the original graph: the line is drawn, and the shading is above the line? Wait, the points above the line: for example, when \(x = 0\), \(y=-1\), and the point \((0,0)\) is above the line (\(y=-1\)), and the shading is blue, so the inequality is \(y\geq-\frac{3}{8}x - 1\)? Wait, no, wait the line: let's re - check the slope. Wait, from \((-8,2)\) to \((0, - 1)\): the change in \(y\) is \(-1 - 2=-3\), change in \(x\) is \(0-(-8)=8\), so slope is \(-\frac{3}{8}\), correct. Y - intercept is \(-1\), correct. Now, the line: is it a solid or dashed line? The graph shows a solid line? Wait, the problem says "linear inequality", if the line is solid, the inequality is \(\geq\) or \(\leq\). Let's take a point in the shaded region, say \((0,0)\). Plug into \(y\) and \(mx + b\): \(0\) vs \(-\frac{3}{8}(0)-1=-1\). Since \(0>-1\), and the line is solid? Wait, no, maybe the line is dashed? Wait, the original graph: the line is drawn with a solid line? Wait, the user's graph: the line is blue, and the shading is above the line. So the inequality is \(y\geq-\frac{3}{8}x - 1\)? Wait, no, wait, let's check again. Wait, the line passes through \((-8,2)\) and \((8, -4)\), and the y - intercept is \((0, - 1)\). Let's write the equation of the line first: \(y=-\frac{3}{8}x - 1\). Now, the shaded region: if the line is solid, and the shading is above the line, then the inequality is \(y\geq-\frac{3}{8}x - 1\). Wait, but let's check the test point \((-8,2)\): plug into \(y\geq-\frac{3}{8}x - 1\): \(2\geq-\frac{3}{8}(-8)-1=3 - 1 = 2\), so \(2\geq2\), which is true. Another point \((8, -4)\): \(-4\geq-\frac{3}{8}(8)-1=-3 - 1=-4\), so \(-4\geq - 4\), which is true. So the inequality is \(y\geq-\frac{3}{8}x - 1\).

Answer:

\(y\geq-\frac{3}{8}x - 1\)