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write the inequality that represents the shaded region

From the graph, we need to identify the equation of the line that serves as the boundary between the shaded and non-shaded regions and then determine whether the inequality is "<" or "≤" (for below the line) or ">" or "≥" (for above the line).

### Step 1: Identifying the line's equation
The line appears to be diagonal, so we can find its equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

By observing the graph:
- The line passes through the points \( (0, 9) \) and \( (3, 0) \).
  
The slope \( m \) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 9}{3 - 0} = \frac{-9}{3} = -3
\]
Thus, the slope \( m = -3 \).

Now, using the point \( (0, 9) \) as the y-intercept \( b \), the equation of the line becomes:
\[
y = -3x + 9
\]

### Step 2: Determining the inequality sign
The shaded region is **below** the line, so the inequality must be either \( y < -3x + 9 \) or \( y \leq -3x + 9 \).

Since the line is solid (not dashed), the inequality includes the boundary, so the correct inequality is:
\[
y \leq -3x + 9
\]

### Final Answer:
The inequality representing the shaded region is:
\[
y \leq -3x + 9
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you determine whether to use "<" or "≤" in an inequality from a graph?
2. How do you calculate the slope of a line given two points?
3. What are the different forms of linear equations and when is each used?
4. How would the inequality change if the shaded region were above the line?
5. How does the appearance of the line (solid or dashed) affect the inequality?

**Tip:** Always check if the line is solid or dashed—this tells you whether to include the line in the inequality (use "≤" or "≥" for solid, and "<" or ">" for dashed).

Use the graph to write the inequality that represents the shaded region.

Learn how to find the inequality representing the shaded region on a graph. This problem involves calculating the slope, identifying the equation of the line in slope-intercept form, and determining the correct inequality. Suitable for students in Grades 8-10.

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