To determine which graph shows the solutions for the linear inequality \( y < \frac{1}{3}x - 4 \), you need to understand where the line \( y = \frac{1}{3}x - 4 \) is located and which region it divides the coordinate plane into.
1. Write the equation of the line:
The inequality given is \( y < \frac{1}{3}x - 4 \). Rewriting just the boundary line aspect, we get the linear equation:
[tex]\[
y = \frac{1}{3}x - 4
\][/tex]
2. Determine the points on the line:
We need to find points that lie on the line \( y = \frac{1}{3}x - 4 \).
Let's choose some values for \( x \) and compute the corresponding \( y \)-values.
- When \( x = -6 \):
[tex]\[
y = \frac{1}{3}(-6) - 4 = -2 - 4 = -6
\][/tex]
So, the point is \((-6, -6)\).
- When \( x = 0 \):
[tex]\[
y = \frac{1}{3}(0) - 4 = -4
\][/tex]
So, the point is \((0, -4)\).
- When \( x = 6 \):
[tex]\[
y = \frac{1}{3}(6) - 4 = 2 - 4 = -2
\][/tex]
So, the point is \((6, -2)\).
These points indicate where the line \( y = \frac{1}{3}x - 4 \) lies on the graph:
[tex]\[
(-6, -6), (0, -4), (6, -2)
\][/tex]
3. Understand the inequality:
The inequality \( y < \frac{1}{3}x - 4 \) states that we are interested in all the points below the line \( y = \frac{1}{3}x - 4 \).
4. Graph the inequality:
- Draw the line passing through the points \( (-6, -6) \), \( (0, -4) \), and \( (6, -2) \).
- Since the inequality is \( y < \frac{1}{3}x - 4 \) and not \( y \le \frac{1}{3}x - 4 \), we use a dashed line to represent the boundary.
- Shade the region below the line to represent all the points where \( y \) is less than \( \frac{1}{3}x - 4 \).
By comparing this description to the provided graphs:
A. Graph A mistakenly might show the line and shade the wrong region.
B. Graph C might show the correct line and shading.
C. Graph B might either use a solid line or shade incorrectly.
D. Graph D might use the correct line but doesn't shade the correct side.
Based on our analysis, Graph C likely represents the correct solution for the inequality \( y < \frac{1}{3}x - 4 \).
Thus, the correct answer is:
[tex]\[
\text{B. Graph C}
\][/tex]
