Question

determine the inequality shown in the graph below. the inequality symbols can be entered using the toolbar.

Explanation:

Step1: Find the slope and equation of the line

The line passes through \((0, -3)\) and \((3, 0)\) (estimated from the graph). The slope \(m=\frac{0 - (-3)}{3 - 0}=1\). Using the slope - intercept form \(y = mx + b\) (where \(b=-3\)), the equation of the line is \(y=x - 3\)? Wait, no, let's check the points again. Wait, when \(x = 0\), \(y=-3\)? Wait, no, looking at the graph, the line passes through \((0,-3)\) and \((3,0)\)? Wait, no, maybe the y - intercept is \(-3\)? Wait, no, let's re - examine. The line passes through \((0, - 3)\) and \((3,0)\)? Wait, no, when \(x = 0\), \(y=-3\), and when \(x = 3\), \(y = 0\). So the slope \(m=\frac{0-(-3)}{3 - 0}=1\). So the equation of the line is \(y=x - 3\)? Wait, no, maybe I made a mistake. Wait, the line in the graph: let's see, when \(x = 0\), \(y=-3\)? Wait, no, looking at the grid, the line passes through \((0,-3)\) and \((3,0)\)? Wait, no, maybe the y - intercept is \(-3\) and the slope is 1. But wait, the line is solid (since the boundary is included, as the line is part of the shaded region? Wait, the line is red and the shaded region is above? Wait, no, the shaded region is above the line? Wait, the line is a straight line with slope 1. Let's find two points on the line. Let's take \(x = 0\), then \(y=-3\)? Wait, no, when \(x = 0\), the line crosses the y - axis at \((0,-3)\)? Wait, no, looking at the graph, the line passes through \((0,-3)\) and \((3,0)\). So the equation of the line is \(y=x - 3\). Now, the shaded region is above the line. Also, the line is solid (since the boundary is included, as the shaded region includes the line). So the inequality is \(y\geq x - 3\). Wait, let's check with a test point. Let's take the origin \((0,0)\). Plug into \(y\) and \(x-3\): \(0\) vs \(0 - 3=-3\). Since \(0\geq - 3\), and the origin is in the shaded region, that works. So the inequality is \(y\geq x - 3\). Wait, but maybe I made a mistake in the y - intercept. Wait, let's re - check. The line passes through \((0,-3)\) and \((3,0)\), so the slope \(m = 1\), equation \(y=x - 3\). The shaded region is above the line, and the line is solid, so the inequality is \(y\geq x - 3\).

Step1: Identify the equation of the boundary line

The boundary line is a straight line. We can find two points on the line. From the graph, the line passes through \((0,-3)\) and \((3,0)\). The slope \(m\) of the line is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-3)}{3 - 0}=\frac{3}{3}=1\). Using the slope - intercept form \(y=mx + b\), where \(m = 1\) and \(b=-3\) (since the line crosses the y - axis at \((0,-3)\)), the equation of the line is \(y=x - 3\).

Step2: Determine the inequality symbol and the direction of the inequality

The line is solid, which means the inequality includes equality (\(\geq\) or \(\leq\)). We choose a test point in the shaded region. Let's use the point \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the expressions \(y\) and \(x - 3\). We get \(y=0\) and \(x-3=0 - 3=-3\). Since \(0\geq - 3\) and \((0,0)\) is in the shaded region, the inequality is \(y\geq x - 3\).

Answer:

\(y\geq x - 3\)