Question

which linear inequality is represented by the graph?
\\( y \leq 2x + 4 \\)
\\( y \leq \frac{1}{2}x + 3 \\)
\\( y \geq \frac{1}{2}x + 3 \\)
\\( y \geq 2x + 3 \\)

Explanation:

Step1: Find the slope and y-intercept of the line

The line passes through \((0, 3)\) (so the y-intercept \(b = 3\)) and another point, say \((2, 4)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 3}{2 - 0}=\frac{1}{2}\). So the equation of the line is \(y=\frac{1}{2}x + 3\).

Step2: Determine the inequality symbol

The shaded region is above the line, and the line is solid (so the inequality includes equality). Thus, the inequality is \(y\geq\frac{1}{2}x + 3\).

Answer:

\(y\geq\frac{1}{2}x + 3\) (corresponding to the option \(y\geq\frac{1}{2}x + 3\))