To determine which graph represents the inequality \( y < \frac{2}{3}x - 2 \), we will follow several steps to understand what the graph should look like and thus identify the correct one.
### Step-by-Step Solution:
1. Rewrite the given inequality:
The inequality given is \( y < \frac{2}{3}x - 2 \).
2. Understand the linear equation of the boundary line:
The boundary line for the inequality \( y < \frac{2}{3}x - 2 \) is:
[tex]\[
y = \frac{2}{3}x - 2
\][/tex]
3. Identify the slope and y-intercept:
In the line equation \( y = \frac{2}{3}x - 2 \):
- The slope (m) is \( \frac{2}{3} \).
- The y-intercept (b) is -2.
4. Plot the boundary line:
- Start by plotting the y-intercept, which is the point (0, -2).
- Use the slope \( \frac{2}{3} \), which means for every 3 units you move to the right on the x-axis, you move 2 units up on the y-axis. From (0, -2), move right 3 units to (3, -2), and then move up 2 units to reach the point (3, 0).
5. Draw the boundary line:
- Connect these points (0, -2) and (3, 0) with a straight line.
- Since the inequality is \( y < \frac{2}{3}x - 2 \), use a dashed line to indicate that points on the line itself are not included in the solution set.
6. Shade the appropriate region:
- For the inequality \( y < \frac{2}{3}x - 2 \), shade the region below the boundary line because we are looking for all points where the y-value is less than \( \frac{2}{3}x - 2 \).
### Conclusion:
Based on these steps, the correct graph will show:
- A dashed line passing through (0, -2) and with a slope that matches \( \frac{2}{3} \).
- The region below this line should be shaded.
By following this detailed explanation, you can identify the correct graph that represents the inequality [tex]\( y < \frac{2}{3}x - 2 \)[/tex].
