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Let's break down the problem and solve it step by step.

### Part a: Write the inequality for the situation

Let:
- \( x \) = number of times Oprah plays Table Tennis.
- \( y \) = number of times Oprah plays Frog Days.

The cost for one game of Table Tennis is 22 tokens, and for one game of Frog Days, it's 55 tokens. Oprah has a total of 3030 tokens.

The total number of tokens used by Oprah can be represented by the equation:
\[
22x + 55y \leq 3030
\]
This inequality represents the situation where Oprah cannot spend more than 3030 tokens.

### Part b: Construct a graph of the region

For this inequality \( 22x + 55y \leq 3030 \), we need to find the boundary line and the region that satisfies this condition.

1. **Boundary Line**: The equality \( 22x + 55y = 3030 \) represents the boundary line.
   - Find the x-intercept: Set \( y = 0 \) in the equation \( 22x + 55y = 3030 \).
   \[
   22x = 3030 \quad \Rightarrow \quad x = \frac{3030}{22} = 137.73
   \]
   Approximate \( x \approx 137 \).

   - Find the y-intercept: Set \( x = 0 \) in the equation \( 22x + 55y = 3030 \).
   \[
   55y = 3030 \quad \Rightarrow \quad y = \frac{3030}{55} = 55.09
   \]
   Approximate \( y \approx 55 \).

2. **Plot the Boundary Line**: Plot the points \( (137, 0) \) and \( (0, 55) \), then draw the line through these points.

3. **Shading the Region**: Since the inequality is \( 22x + 55y \leq 3030 \), the region below or on the line represents all possible combinations of \( x \) and \( y \) where Oprah spends 3030 tokens or fewer.

### Part c: Interpret the x- and y-intercepts

- **x-intercept (137, 0)**: This means Oprah could play Table Tennis 137 times and spend all 3030 tokens without playing any Frog Days games.
- **y-intercept (0, 55)**: This means Oprah could play Frog Days 55 times and spend all 3030 tokens without playing any Table Tennis games.

Would you like a more detailed explanation or further help with the graph? Here are 5 related questions that may deepen your understanding:

1. What happens to the inequality if the total tokens are increased or decreased?
2. How would the graph change if the cost per game for Table Tennis increased?
3. What does the slope of the boundary line represent in this context?
4. How would you graph the region if there were constraints on how many times Oprah could play each game (e.g., a maximum of 100 times)?
5. How can you verify the solution set for the inequality graphically?

**Tip:** Always check the boundary points when graphing inequalities, as they help in understanding the feasible region effectively.

Oprah is thinking about how to use her last arcade tokens. A game of Table Tennis costs 22 tokens and a game of Frog Days costs 55 tokens. She has a total of 3030 tokens left. Write the inequality to represent this situation and graph the solution region.

Explore how to solve a linear inequality involving the cost of two arcade games and a token budget. The problem covers graphing the inequality 22x + 55y ≤ 3030, interpreting the x- and y-intercepts, and constructing a feasible region for different gameplay combinations.

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