https://file.aiquickdraw.com/mathbot/file/1727195080282brxcn56s.jpg

The graph shown in the image represents a linear inequality. We need to determine which inequality is correctly depicted by the shaded region and the line on the graph. Here's how to analyze it:

1. **Identify the line**: The equation of the line can be identified by determining its slope and y-intercept.
2. **Slope determination**: The line appears to pass through the points (0,1) and (4,0). To find the slope \( m \) of the line, use the formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
3. **Y-intercept**: The y-intercept can be read directly as the point where the line crosses the y-axis, which is \( y = 1 \).

After calculating these values, we can confirm which inequality option is correct by analyzing the shading relative to the line. Let's perform these calculations.The equation of the line is \( y = -0.25x + 1 \), which simplifies to \( y = -\frac{1}{4}x + 1 \).

From the graph, we observe that the shaded region is above the line, indicating that the inequality should include values of \( y \) that are greater than or equal to the line's values. Therefore, the correct inequality representation of the graph is:

\[ y \geq -\frac{1}{4}x + 1 \]

Would you like further details on how to analyze inequalities on a graph or have any other questions?

Here are five related questions you might find interesting:
1. What happens to the graph of an inequality if we change \( \geq \) to \( > \)?
2. How would the graph change if the slope were positive instead of negative?
3. What steps would you follow to graph the inequality \( y < 2x - 5 \)?
4. How do you determine the region to shade for the inequality \( y \leq 3 \)?
5. What is the significance of using a dashed line versus a solid line in graphing inequalities?

**Tip**: Always double-check the points on the graph and their corresponding values in the inequality to ensure the shading is on the correct side of the line.

Which inequality has the graph shown below?

Explore how to graph the linear inequality y ≥ -1/4x + 1. Learn about slope determination, equation of a line, and shading regions to solve inequalities. Suitable for students in Grades 8-10.

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