Question

which linear inequality is represented by the graph?
option 1: $y > \frac{1}{3}x - 4$
option 2: $y \leq \frac{1}{3}x - 4$
option 3: $y \leq \frac{1}{3}x + 4$
option 4: $y \geq \frac{1}{3}x + 4$

Explanation:

Step1: Identify the slope and y-intercept

The line in the graph has a slope \( m=\frac{1}{3} \) (since for every 3 units right, it goes up 1 unit) and y-intercept \( b = - 4\)? Wait, no, looking at the graph, when \( x = 0\), the line crosses at \( y=-4\)? Wait, no, wait the options have \( + 4\) or \( - 4\). Wait, let's re - examine. Wait, the shaded region: first, the line equation is in slope - intercept form \( y=mx + b\). Let's find two points on the line. From the graph, when \( x = 0\), \( y=-4\)? No, wait the options: let's check the y - intercept. Wait, the line passes through (0, - 4)? Wait, no, maybe I made a mistake. Wait, the options are \( y\geq\frac{1}{3}x - 4\), \( y\leq\frac{1}{3}x - 4\), \( y\leq\frac{1}{3}x + 4\), \( y\geq\frac{1}{3}x + 4\). Wait, let's find the slope. The slope \( m=\frac{1}{3}\) (rise over run: 1 up for 3 right). Now, the y - intercept: when \( x = 0\), the line is at \( y=-4\)? Wait, no, the shaded region: if the line is \( y=\frac{1}{3}x - 4\), and the shaded region is above the line (since the line is solid and the shading is above), then the inequality is \( y\geq\frac{1}{3}x - 4\)? Wait, no, wait the graph: let's see the y - intercept. Wait, maybe I messed up. Wait, the line: let's take two points. Let's say when \( x = 3\), \( y=-3\) (since slope is \( \frac{1}{3}\), from \( x = 0,y=-4\), then \( x = 3,y=-4 + 1=-3\)). And when \( x = 0,y=-4\). Now, the shaded region: if the line is solid and the shading is above the line, then the inequality is \( y\geq\frac{1}{3}x - 4\). Wait, but let's check the options. Wait, the first option is \( y\geq\frac{1}{3}x - 4\), second is \( y\leq\frac{1}{3}x - 4\), third \( y\leq\frac{1}{3}x + 4\), fourth \( y\geq\frac{1}{3}x + 4\). Wait, maybe I made a mistake in y - intercept. Wait, maybe the y - intercept is - 4. So the line is \( y=\frac{1}{3}x - 4\), and the shaded region is above the line (since the line is solid, and the shading is above), so the inequality is \( y\geq\frac{1}{3}x - 4\). Wait, but let's confirm. The general form of a linear inequality: if the line is solid, the inequality is \(\geq\) or \(\leq\). If the shading is above the line, it's \( y\geq mx + b\); if below, \( y\leq mx + b\). So if the line is \( y=\frac{1}{3}x - 4\) and shading is above, then \( y\geq\frac{1}{3}x - 4\).

Step2: Check the inequality direction

The line is solid, so the inequality includes equality. The shaded region is above the line \( y = \frac{1}{3}x-4\), so the inequality is \( y\geq\frac{1}{3}x - 4\).

Answer:

\( y\geq\frac{1}{3}x - 4\) (the first option: \( y\geq\frac{1}{3}x - 4\))