Question

linear equations and inequalities
which linear inequality is represented by the graph?
(3, -0.3)
(0, -1.3)
$y \geq \frac{1}{3}x - \frac{4}{3}$
$y \leq \frac{1}{3}x - \frac{4}{3}$
$y \geq \frac{1}{3}x - 1.3$
$y \leq \frac{1}{3}x - 1.3$

Explanation:

Step1: Determine the slope

The two points on the line are \((0, -1.3)\) and \((3, -0.3)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-0.3-(-1.3)}{3 - 0}=\frac{1}{3}\).

Step2: Determine the y - intercept

The line crosses the y - axis at \((0,-1.3)\), so the y - intercept \(b=-1.3 =-\frac{4}{3}\) (since \(-1.3\approx-\frac{4}{3}\)).

Step3: Determine the inequality symbol

The shaded region is below the line, and the line is solid (so the inequality includes equality). So the inequality is \(y\leq\frac{1}{3}x - \frac{4}{3}\) (or \(y\leq\frac{1}{3}x-1.3\), but \(-\frac{4}{3}\approx - 1.33\) and \(-1.3\) is an approximation, the exact fraction form is better here). Looking at the options, the second option \(y\leq\frac{1}{3}x-\frac{4}{3}\) matches.

Answer:

\(y\leq\frac{1}{3}x - \frac{4}{3}\) (the second option in the list of given options)