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?

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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).