https://file.aiquickdraw.com/mathbot/file/1730066656506hxx1yai1.jpg

To match each shaded region in the plots with the corresponding inequality, let’s analyze each plot and inequality.

### Steps to Solve
1. **Identify the boundary line of the shaded area in each plot** to understand which inequality might represent it.
2. **Check if the shading is above or below the line** for inequalities involving y, or to the left or right for inequalities involving x.
3. **Match the characteristics of each inequality** with the graphs provided.

### Inequality Analysis
- **(a)** \( x \geq y - 2 \): Rearranged to \( y \leq x + 2 \), shading is below the line.
- **(b)** \( y + 2x \leq 1 \): Rearranged to \( y \leq -2x + 1 \), shading below a line with a slope of -2.
- **(c)** \( x \geq y + 2 \): Rearranged to \( y \leq x - 2 \), shading below the line.
- **(d)** \( 2y + x \leq 1 \): Rearranged to \( y \leq -\frac{1}{2}x + \frac{1}{2} \).
- **(e)** \( y \geq x - 2 \): Shading is above the line with slope 1.
- **(f)** \( y \leq x + 2 \): Shading below the line.
- **(g)** \( y + 2x \geq 1 \): Rearranged to \( y \geq -2x + 1 \), shading above the line with slope -2.

### Matching Each Plot
1. **Top Plot (a)**: Matches inequality **(g) \( y + 2x \geq 1 \)**, since the shading is above the line with a slope of -2.
2. **Middle Plot (e)**: Matches inequality **(e) \( y \geq x - 2 \)**, since the shading is above the line with slope 1.
3. **Bottom Plot (d)**: Matches inequality **(d) \( 2y + x \leq 1 \)**, since the shading is below the line with a slope of -\(\frac{1}{2}\).

### Final Answer
- **Top Plot (a): g. \( y + 2x \geq 1 \)**
- **Middle Plot (e): e. \( y \geq x - 2 \)**
- **Bottom Plot (d): d. \( 2y + x \leq 1 \)**

Would you like a detailed explanation of how each inequality was matched to its plot or have any other questions?

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#### Related Questions:
1. How do you rearrange inequalities to better match graphing?
2. How can you identify if shading is above or below a line?
3. What is the significance of slope in identifying inequalities?
4. Why are inequalities reversed when multiplying by a negative number?
5. How do boundary lines impact the shading of inequalities?

#### Tip:
When graphing inequalities, always check if the line should be solid (for \( \geq \) or \( \leq \)) or dashed (for \( > \) or \( < \)).

Match each shaded region in the plots with the corresponding inequality from the given options.

Learn how to match shaded regions on graphs with their corresponding inequalities by analyzing the slope, shading direction, and boundary line properties. Ideal for students in Grades 9-10 studying algebra and graphing inequalities.

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