Question

Question was provided via image upload.

Explanation:

Step1: Find two points on the line

The line passes through \((0, -3)\) and \((4, 0)\).

Step2: Calculate the slope \(m\)

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), substitute \((x_1, y_1)=(0, -3)\) and \((x_2, y_2)=(4, 0)\):
\(m=\frac{0 - (-3)}{4 - 0}=\frac{3}{4}\)

Step3: Determine the inequality

The line is \(y=\frac{3}{4}x - 3\). The shaded region is above the line (since the blue area is above the line), and the line is solid (wait, no, the line is blue, but the shading: looking at the graph, the shaded area is where \(y\geq\frac{3}{4}x - 3\)? Wait, no, let's check the y-intercept. The line crosses the y-axis at \((0, -3)\) and x-axis at \((4, 0)\). The slope is \(\frac{3}{4}\), so the equation is \(y=\frac{3}{4}x - 3\). The shaded region is above the line? Wait, no, when \(x = 0\), the shaded area is above \(y=-3\)? Wait, the graph shows the blue area is to the left of the line? Wait, no, the line is increasing, and the shaded area is where \(y\geq\frac{3}{4}x - 3\)? Wait, let's test a point in the shaded area, say \((0, 0)\). Plug into \(y\) and \(\frac{3}{4}x - 3\): \(0\) vs \(\frac{3}{4}(0)-3=-3\). Since \(0\geq - 3\), so the inequality is \(y\geq\frac{3}{4}x - 3\), or \(4y\geq3x - 12\), or \(3x - 4y\leq12\). Wait, maybe the question is to find the inequality. Let's re - express:

From the two points \((0, -3)\) and \((4, 0)\), the slope \(m=\frac{0 - (-3)}{4 - 0}=\frac{3}{4}\). The equation of the line is \(y=\frac{3}{4}x - 3\). The shaded region is above the line (since at \(x = 0\), the shaded area includes \(y = 0\) which is above \(y=-3\)), and the line is solid (wait, the line is blue, and the shading is on one side). So the inequality is \(y\geq\frac{3}{4}x - 3\), multiplying both sides by 4: \(4y\geq3x - 12\), or \(3x - 4y\leq12\).

Wait, maybe the problem is to write the inequality represented by the graph. Let's confirm:

1. Find the equation of the line:

• Two points: \((0, -3)\) (y - intercept) and \((4, 0)\) (x - intercept).
• Slope \(m=\frac{0 - (-3)}{4 - 0}=\frac{3}{4}\).
• Equation: \(y=\frac{3}{4}x - 3\).

1. Determine the inequality sign:

• The shaded region is above the line (test point \((0,0)\): \(0\geq\frac{3}{4}(0)-3=-3\), which is true). Also, the line is solid (so the inequality is \(\geq\) or \(\leq\); since the shaded area is above, it's \(y\geq\frac{3}{4}x - 3\)).
• To write in standard form: Multiply both sides by 4: \(4y\geq3x - 12\), then \(3x - 4y\leq12\).

Answer:

The inequality represented by the graph is \(y\geq\frac{3}{4}x - 3\) (or \(3x - 4y\leq12\))