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 the points \((-8, -2)\) and \((0, 6)\). The slope \(m\) is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - (-2)}{0 - (-8)}=\frac{8}{8} = 1\).

Step2: Determine the y - intercept

The y - intercept \(b\) is the value of \(y\) when \(x = 0\). From the point \((0,6)\), we know that \(b = 6\). So the equation of the line in slope - intercept form (\(y=mx + b\)) is \(y=x + 6\).

Step3: Determine the inequality symbol

The line is dashed (wait, no, looking at the graph, the line is a solid line? Wait, the shaded region is above the line? Wait, let's check the graph. The line passes through \((-8,-2)\) and \((0,6)\), and the shaded region is above the line. Also, since the line is a solid line (the boundary is included), the inequality symbol is \(\geq\). Wait, let's verify with a test point. Let's take the point \((0,7)\) which is in the shaded region. Plug into \(y=x + 6\): \(7=0 + 6\)? No, \(7>0 + 6\). Wait, maybe I made a mistake in the slope. Wait, let's recalculate the slope. Let's take two points: when \(x=-6\), \(y = 0\) (from the x - intercept) and when \(x = 0\), \(y=6\). So the slope \(m=\frac{6-0}{0 - (-6)}=\frac{6}{6}=1\). So the line is \(y=x + 6\). Now, let's take a point in the shaded area, say \((0,7)\). Substitute into \(y\) and \(x + 6\): \(7\) and \(0 + 6=6\). Since \(7>6\), and the line is a solid line (so the boundary is included), the inequality is \(y\geq x + 6\)? Wait, no, wait the graph: when \(x=-8\), \(y=-2\), and the shaded region is above the line. Wait, let's check the point \((-8,-2)\) on the line. If we take a point below the line, say \((-8,-3)\), \(y=-3\), and \(x + 6=-8 + 6=-2\). Since \(-3<-2\), and the shaded region is above the line, and the line is solid (so the inequality is \(\geq\)). Wait, but let's check the graph again. The line goes from \((-8,-2)\) to \((0,6)\) to \((8,14)\) (extending). The shaded area is above the line. So the inequality is \(y\geq x+6\)? Wait, no, wait when \(x = 0\), \(y = 6\), and the shaded area is above, so for \(x = 0\), \(y\) is greater than or equal to \(6\). So the inequality is \(y\geq x + 6\). Wait, but let's check the slope again. Wait, another way: the standard form of a line is \(y=mx + b\). We found \(m = 1\), \(b = 6\), and since the shaded region is above the line and the line is solid, the inequality is \(y\geq x+6\).

Wait, maybe I made a mistake in the slope. Let's take two points: \((-6,0)\) (x - intercept) and \((0,6)\). Slope \(m=\frac{6 - 0}{0-(-6)} = 1\). So the line is \(y=x + 6\). The shaded region is above the line, and the line is solid (so the inequality is \(\geq\)). So the inequality in slope - intercept form is \(y\geq x + 6\).

Answer:

\(y\geq x + 6\)