Question

graphing two - variable linear inequalities
which linear inequality is represented by the graph?
$y\geq \frac{1}{2}x + 2$
$y\geq \frac{1}{3}x + 2$
$y\leq \frac{1}{2}x + 2$
$y\leq \frac{1}{3}x + 2$

Explanation:

Step1: Find the slope of the line

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

Step2: Determine the inequality symbol

The line is solid, so the inequality includes equality. The shaded region is above the line (since when \(x = 0\), \(y\geq2\) which matches the \(y\)-intercept \(2\) and the 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:

\(y\geq\frac{1}{2}x + 2\) (the first option: \(y\geq\frac{1}{2}x+2\))