Question

which linear inequality is represented by the graph?
options:
$y \leq \frac{1}{2}x + 2$
$y \geq \frac{1}{2}x + 2$
$y \leq \frac{1}{3}x + 2$
$y \geq \frac{1}{3}x + 2$
(graph: a coordinate plane with a line passing through (0,2) and (4,4), shaded above the line, solid line)

Explanation:

Step1: Find the slope of the line

The line passes through \((0, 2)\) and \((4, 4)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{4 - 0}=\frac{2}{4}=\frac{1}{2}\).

Step2: Determine the inequality symbol

The line is solid, so the inequality includes equality. The shaded region is above the line, so the inequality is \(y\geq\) the line equation. The line equation is \(y = \frac{1}{2}x + 2\) (using slope - intercept form \(y=mx + b\) with \(m=\frac{1}{2}\) and \(b = 2\)). So the inequality is \(y\geq\frac{1}{2}x + 2\).

Answer:

B. \(y\geq\frac{1}{2}x + 2\)