https://file.aiquickdraw.com/mathbot/file/1730127441790bee2amsp.jpg

The question provides the following system of inequalities:

\[
\begin{cases}
x \geq -2 \\
x - 2y - 2 < 0
\end{cases}
\]

We need to determine which graph correctly represents the solution set for this system.

### Step 1: Analyzing Each Inequality
1. **First Inequality**: \( x \geq -2 \)
   - This inequality represents a vertical line at \( x = -2 \) with shading to the right (where \( x \) values are greater than or equal to \(-2\)).
   
2. **Second Inequality**: \( x - 2y - 2 < 0 \)
   - Rearrange to solve for \( y \):
     \[
     x - 2y - 2 < 0
     \]
     \[
     -2y < -x + 2
     \]
     \[
     y > \frac{1}{2}x - 1
     \]
   - This inequality represents a line with slope \( \frac{1}{2} \) and y-intercept \(-1\), with shading above the line (since \( y \) is greater than the line).

### Step 2: Identifying the Solution
We are looking for a graph where:
- There is a vertical boundary line at \( x = -2 \) with shading to the right.
- There is an oblique boundary line with a slope of \( \frac{1}{2} \) and y-intercept at \(-1\), with shading above this line.

### Solution
Upon examining the options in the image, **Graph C** correctly shows:
- A shaded region to the right of \( x = -2 \).
- A shaded region above the line \( y = \frac{1}{2}x - 1 \).

Thus, the correct answer is **Graph C**.

Would you like a deeper explanation, or do you have any questions?

### Related Questions
1. How would the solution set change if the inequality \( x \geq -2 \) was instead \( x > -2 \)?
2. What would be the solution set if the inequality \( y > \frac{1}{2}x - 1 \) was changed to \( y \leq \frac{1}{2}x - 1 \)?
3. How can you determine the slope and intercept of a line given in a general inequality form?
4. What is the difference between strict inequalities and inclusive inequalities in graphing?
5. How does the shading direction differ between \( y > \) and \( y < \) inequalities?

### Tip
When graphing inequalities, always identify boundary lines and carefully determine shading based on whether the inequality is strict (\( < \) or \( > \)) or inclusive (\( \leq \) or \( \geq \)).

Which graph shows the solution set of the system of inequalities: x ≥ -2 and x - 2y - 2 < 0?

Learn how to graph the solution set of the linear inequalities x ≥ -2 and x - 2y - 2 < 0. This problem demonstrates key algebraic concepts such as identifying shading regions and rearranging inequalities for slope-intercept form. Suitable for students in Grades 9-10.

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