To determine which linear inequality is represented on the graph, let's analyze the key characteristics of the shaded region and the boundary line.

Step 1: Analyzing the Slope and Y-Intercept

1. The boundary line on the graph has a slope and intercept we can identify:

   • The line appears to pass through points such as $(0, -6)$ and $(2, -5)$.
   • Using these points, we can calculate the slope $m$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 + 6}{2 - 0} = \frac{1}{2}$
   • Therefore, the slope of the line is $\frac{1}{2}$, and the y-intercept is $-6$.

2. Thus, the equation of the line is: $y = \frac{1}{2}x - 6$

Step 2: Determining the Inequality

1. The region below the line is shaded, indicating a "less than or equal to" inequality.
2. Since the line is dashed, it suggests a strict inequality (either $<$ or $>$).

Conclusion

The inequality that represents the shaded region is: $y < \frac{1}{2}x - 6$

So, the correct answer is: $y < \frac{1}{2}x - 6$.

Would you like a breakdown of how to interpret different types of inequalities on graphs?

Here are some follow-up questions that may help deepen your understanding:

1. How do you determine the slope of a line from a graph?
2. Why does a dashed line represent a strict inequality?
3. What does shading below the line indicate in terms of inequalities?
4. How can you determine if a graph shows a "less than" or "greater than" inequality?
5. What would change if the line were solid instead of dashed?

Tip: Remember, when identifying inequalities from graphs, the slope and y-intercept of the line, along with the shading direction, are key indicators of the inequality type.