Question

write an inequality that represents the graph.

Explanation:

Step1: Find the slope-intercept form of the boundary line

First, identify two points on the boundary line. From the graph, we can see the line passes through \((-4, 2)\) and \((0, -2)\) (assuming the y-intercept is at \((0, -2)\) and another point is \((-4, 2)\)). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2 - 2}{0 - (-4)}=\frac{-4}{4}=-1\). The y-intercept \(b\) is \(-2\), so the equation of the line in slope - intercept form \(y = mx + b\) is \(y=-x - 2\).

Step2: Determine the inequality symbol

The boundary line is solid or dashed? Since the line in the graph appears to be solid (from the visual, the shading and the line), but wait, actually, looking at the shading, the shaded region is below or above? Wait, let's re - check the points. Wait, another way: let's take a test point. Let's take the point \((-5,0)\) which is in the shaded region. Plug into the line equation \(y=-x - 2\). Left - hand side \(y = 0\), right - hand side \(-(-5)-2=5 - 2 = 3\). Since \(0\leq3\)? Wait, no, wait, maybe the line is \(y=-x - 2\), and the shading is below the line? Wait, let's re - calculate the slope. Wait, maybe the two points are \((-4,2)\) and \((0, - 2)\) as before. The slope \(m=-1\), equation \(y=-x - 2\). Now, take the test point \((-5,0)\): substitute into \(y\) and \(-x - 2\). \(y = 0\), \(-x - 2=-(-5)-2 = 3\). Since \(0<3\)? Wait, no, maybe the line is \(y=-x - 2\), and the inequality is \(y\leq - x - 2\)? Wait, no, let's check the direction of the shading. Wait, the shaded region seems to be where \(y\leq - x - 2\)? Wait, no, let's re - examine the graph. Wait, the line goes from the top left (around \((-4,2)\)) to the bottom right (around \((0, - 2)\)). The shaded area is to the left and below? Wait, maybe I made a mistake in the points. Let's try another approach. Let's find two points on the line: when \(x=-4\), \(y = 2\); when \(x=-2\), \(y = 0\) (since when \(x=-2\), \(y=-(-2)-2=0\)). So the line is \(y=-x - 2\). Now, the shading: let's take a point in the shaded area, say \((-5, - 3)\). Plug into \(y\) and \(-x - 2\): \(y=-3\), \(-x - 2=-(-5)-2 = 3\). Since \(-3\leq3\), and the line is solid (so the inequality includes equality), so the inequality is \(y\leq - x - 2\), or \(x + y\leq - 2\). Wait, maybe the correct two points are \((-4,2)\) and \((0, - 2)\). Let's re - calculate the slope: \(m=\frac{-2 - 2}{0+4}=\frac{-4}{4}=-1\), equation \(y=-x - 2\). The shaded region is below the line (since the test point \((-5,0)\) gives \(0\leq -(-5)-2 = 3\), so \(y\leq -x - 2\)).

Wait, but maybe the line is \(y=-x - 2\), and the inequality is \(y\leq -x - 2\) or \(x + y\leq - 2\). Let's verify with another point. Take \((-3,0)\): \(y = 0\), \(-x - 2=-(-3)-2 = 1\), \(0\leq1\), which is true, and \((-3,0)\) is on the boundary or in the shaded region?

Step3: Write the final inequality

After determining the equation of the line \(y=-x - 2\) and the inequality symbol (since the shaded region is below the line and the line is solid, the inequality is \(y\leq -x - 2\) or \(x + y\leq - 2\)).

Wait, maybe I made a mistake in the points. Let's re - look at the graph. The line passes through \((-4,2)\) and \((0, - 2)\), so the equation is \(y=-x - 2\). The shaded area is where \(y\leq -x - 2\) (because when we take a point in the shaded area, say \((-5, - 5)\), \(y=-5\), \(-x - 2=-(-5)-2 = 3\), and \(-5\leq3\)).

Answer:

The inequality that represents the graph is \(y\leq -x - 2\) (or \(x + y\leq - 2\))