Mr. Benjamin gave his students the following system of linear inequalities
 

x ≥ −2x ≥ −2

x − 2y − 2 <0x − 2y − 2 <0


Which graph shows the solution set of this system of inequalities?

A.

B.

C.

D.

**4b.** 

Which point is part of the solution set?

A.(8, 3)

B.(‒3, 5)

C.(‒2, ‒8)

D.(10, 7)


To solve this system of inequalities, let's examine each inequality individually to determine the solution region.

### System of Inequalities
The inequalities given are:
1. \( x \geq -2 \)
2. \( x - 2y - 2 < 0 \)

### Step 1: Graphing the Inequalities

1. **First Inequality: \( x \geq -2 \)**
   - This inequality represents a vertical line at \( x = -2 \).
   - The solution includes all points to the right of this line.

2. **Second Inequality: \( x - 2y - 2 < 0 \)**
   - Rearrange this inequality to get:
     \[
     x - 2y < 2
     \]
     or
     \[
     2y > x - 2 \Rightarrow y > \frac{x - 2}{2}
     \]
   - This inequality represents a line with a slope of \( \frac{1}{2} \) and a y-intercept of \( -1 \).
   - Since we have \( y > \frac{x - 2}{2} \), the solution region is above this line.

### Step 2: Solution Set of the System
The solution set will be the region where both conditions are satisfied:
1. Points to the right of \( x = -2 \).
2. Points above the line \( y = \frac{x - 2}{2} \).

### Step 3: Determining Which Point is Part of the Solution Set

Let's test each point in the options:

- **A. (8, 3)**
  - Check \( x \geq -2 \): \( 8 \geq -2 \) (True)
  - Check \( y > \frac{x - 2}{2} \): \( 3 > \frac{8 - 2}{2} = 3 \) (False)
  - This point is **not** part of the solution.

- **B. (‒3, 5)**
  - Check \( x \geq -2 \): \( -3 \geq -2 \) (False)
  - This point is **not** part of the solution.

- **C. (‒2, ‒8)**
  - Check \( x \geq -2 \): \( -2 \geq -2 \) (True)
  - Check \( y > \frac{x - 2}{2} \): \( -8 > \frac{-2 - 2}{2} = -2 \) (False)
  - This point is **not** part of the solution.

- **D. (10, 7)**
  - Check \( x \geq -2 \): \( 10 \geq -2 \) (True)
  - Check \( y > \frac{x - 2}{2} \): \( 7 > \frac{10 - 2}{2} = 4 \) (True)
  - This point **is** part of the solution.

### Answers
1. **Graph Solution:** Based on the information, look for the graph showing the region to the right of \( x = -2 \) and above the line \( y = \frac{x - 2}{2} \).
2. **Point Solution:** The answer for point inclusion is **D. (10, 7)**.

Would you like further details on each step, or have any questions about graphing linear inequalities?

---

1. What are some key indicators to identify the solution set of a system of inequalities?
2. How can you verify if a point lies within a region defined by inequalities?
3. Why is it important to test each point in multiple inequalities for a system?
4. How does rearranging an inequality help in graphing?
5. What strategies are helpful for visualizing solutions of inequalities on a coordinate plane?

**Tip:** When graphing inequalities, always check whether the inequality sign is strict (< or >) or inclusive (≤ or ≥), as it affects the line style (dotted or solid).

Mr. Benjamin gave his students the following system of linear inequalities: x ≥ −2 and x − 2y − 2 < 0. Which graph shows the solution set of this system of inequalities? Also, which point is part of the solution set?
Options:
For the graph:
A, B, C, D.
For the point:
A. (8, 3)
B. (‒3, 5)
C. (‒2, ‒8)
D. (10, 7)

Explore how to determine the solution set for the system of linear inequalities x ≥ −2 and x − 2y − 2 < 0. This solution covers graphing inequalities on the coordinate plane and identifying valid solution points, suitable for students in Grades 9-10.

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