Question

which linear inequality is represented by the graph?
$\bigcirc$ $y \leq \frac{1}{3}x - 1$
$\bigcirc$ $y \geq \frac{1}{3}x - 1$
$\bigcirc$ $y < 3x - 1$
$\bigcirc$ $y > 3x - 1$

Explanation:

Step1: Determine the slope and y-intercept

The line passes through points \((0, -1)\) (y-intercept) and \((3, 0)\). The slope \(m=\frac{0 - (-1)}{3 - 0}=\frac{1}{3}\). So the equation of the line is \(y = \frac{1}{3}x-1\).

Step2: Determine the inequality symbol

The line is solid (so the inequality is \(\leq\) or \(\geq\)) and the shaded region is below the line (so \(y\leq\) the line equation). So the inequality is \(y\leq\frac{1}{3}x - 1\).

Answer:

\(y\leq\frac{1}{3}x - 1\) (corresponding to the first option: \(y\leq\frac{1}{3}x - 1\))