Question

which linear inequality is represented by the graph? options: y < (2/3)x + 3; y > (3/2)x + 3; y > (2/3)x + 3; y < (3/2)x + 3

Explanation:

Step1: Determine the slope of the line

The line passes through the points \((0, 3)\) and \((3, 5)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 3}{3 - 0}=\frac{2}{3}\).

Step2: Determine the inequality sign

The line is dashed, so the inequality is either \(>\) or \(<\). The shaded region is above the line, so the inequality sign is \(>\).

Step3: Write the equation of the line

Using the slope - intercept form \(y = mx + b\) where \(m=\frac{2}{3}\) and \(b = 3\) (the y - intercept), the equation of the line is \(y=\frac{2}{3}x + 3\). Combining with the inequality sign from Step 2, the inequality is \(y>\frac{2}{3}x + 3\).

Answer:

\(y>\frac{2}{3}x + 3\) (the third option: \(y>\frac{2}{3}x + 3\))