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The problem involves comparing the cost of two companies, $ P $ and $ R $, for ordering trophies. Let's break it down:

- **Company P**: 
  - Charges $3.50 per trophy.
  - Has a one-time engraving fee of $25.
  - Total cost for $ x $ trophies: $ 3.5x + 25 $.

- **Company R**: 
  - Charges $7.50 per trophy.
  - Has a one-time engraving fee of $17.
  - Total cost for $ x $ trophies: $ 7.5x + 17 $.

The inequality needs to determine when the total charge of Company P is less than the total charge of Company R.

We want the inequality to be:
\[
3.5x + 25 < 7.5x + 17
\]
Now, solving this inequality:

1. Subtract $ 3.5x $ from both sides:
   \[
   25 < 4x + 17
   \]

2. Subtract 17 from both sides:
   \[
   8 < 4x
   \]

3. Divide both sides by 4:
   \[
   2 < x
   \]

This means $ x > 2 $, so the minimum number of trophies $ x $ must be greater than 2 for Company P to charge less than Company R.

Thus, the correct inequality is:
\[
3.5x + 25 < 7.5x + 17
\]

The correct answer is the third option:
**$ 3.5x + 25 < 7.5x + 17 $**.

Would you like further details or have any questions?

Here are 5 related questions you can think about:
1. What happens if the engraving fees are removed from both companies' costs?
2. How would the inequality change if Company P increased its per trophy charge?
3. How do we solve inequalities when variables are present on both sides?
4. How does the one-time fee affect the decision for small orders compared to larger orders?
5. What is the graphical representation of this inequality?

**Tip:** When solving inequalities, treat them similarly to equations, but remember to reverse the inequality sign if you multiply or divide by a negative number.

Veronica is ordering trophies for her school. Company P charges $3.50 for each trophy and a one-time engraving fee of $25. Company R charges $7.50 for each trophy and a one-time engraving fee of $17. Which inequality can be used to find x, the minimum number of trophies that can be ordered so that the total charge at Company P is less than the total charge at Company R?

Learn how to solve the inequality for a cost comparison between two companies offering trophies. Find out the minimum number of trophies required to make Company P's total charge less than Company R's. This problem involves linear inequalities and is suitable for Grades 7-9.

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