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Let's analyze the given problem and its conditions:

- A merchant buys a certain number of pencils for 180,000 pesos and sells all except 6 with a profit of 2,000 pesos per pencil.
- With the money from the sales, the merchant could have bought 30 more pencils than initially.
  
We need to determine which statements (I, II, and III) are correct:

1. **I. The equation for the money collected is \( 2(x - 6) \):**
   - Let \( x \) be the number of pencils the merchant initially bought.
   - The merchant sells \( x - 6 \) pencils with a profit of 2,000 pesos per pencil.
   - The total profit would be \( 2,000(x - 6) \), which simplifies to \( 2(x - 6) \) in terms of thousands of pesos. This matches the equation given in statement I.
   - **Statement I is true.**

2. **II. The merchant bought 42 pencils:**
   - The merchant uses the money earned from sales to buy 30 more pencils. Therefore, the total value of the pencils bought after selling is:
     \[
     180,000 + 2,000(x - 6) = (x + 30) \cdot \text{price per pencil}.
     \]
     Solving this would help verify if 42 pencils were bought.
   - For now, we don't have enough information to immediately confirm this, but it will be evaluated during calculations.

3. **III. The cost of each pencil is 6.29 thousand pesos:**
   - This seems to suggest the price of one pencil is 6,290 pesos. We'll need to calculate the exact cost per pencil based on the available information and confirm this statement.

Let me now calculate the value for \( x \) (the number of pencils).The equation gives two potential solutions for \( x \):

\[
x = 3 - 3\sqrt{301} \quad \text{(negative value, not valid)}
\]
\[
x = 3 + 3\sqrt{301} \quad \text{(positive value, valid)}
\]

Approximating \( x \), the valid solution represents the number of pencils bought. Let's compute the approximate value of \( x \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

In the meantime, based on the symbolic result, it appears that the number of pencils the merchant bought can be calculated from \( x = 3 + 3\sqrt{301} \), which would need to be approximated.

Would you like me to guide you through calculating this manually, or do you have any other questions related to the problem?

Here are 5 relative questions to explore further:
1. How would you approximate \( \sqrt{301} \)?
2. How does the equation \( 180,000 + 2000(x-6) \) represent the profit?
3. What steps would be involved in checking if the second statement is true?
4. Can we derive the price per pencil directly from the cost equation?
5. What other similar problems can be solved using this approach?

**Tip:** When dealing with quadratic-like equations involving square roots, it's helpful to approximate square root values manually if a calculator isn't handy.

A merchant buys a certain number of pencils for 180,000 pesos and sells all except 6 with a profit of 2,000 pesos per pencil. Knowing that with the money collected from the sale, he could have bought 30 more pencils than before, determine the price of each pencil. Which of the following statements is true?
I. The equation that represents the money collected is 2(x-6).
II. He bought 42 pencils.
III. The cost of one pencil is 6.29 thousand pesos.

Solve a linear equation problem where a merchant buys pencils for 180,000 pesos and sells them with a 2,000-peso profit per pencil. Explore how to determine the cost of each pencil and the validity of given statements. Ideal for students in Grades 10-12.

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