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The problem presented asks for a counterexample that can verify whether the conjecture \(5x + 6 > 6x\) is false for any \(x\). To solve this, we need to explore the given inequality and check values of \(x\).

### Step-by-Step Analysis:
The inequality is:

\[
5x + 6 > 6x
\]

1. **Simplify the inequality:**
   Subtract \(5x\) from both sides:

   \[
   6 > x
   \]

2. **Interpret the result:**
   The inequality simplifies to \(6 > x\), meaning that the conjecture \(5x + 6 > 6x\) holds only when \(x\) is less than 6. If \(x\) is equal to or greater than 6, the inequality is false.

### Verifying the options:
Now let's check the values of \(x\) provided to see which one acts as a counterexample:

- \(x = 0\): 
  \[
  5(0) + 6 = 6, \quad 6(0) = 0
  \]
  \[
  6 > 0 \quad \text{(True)}
  \]

- \(x = 2\): 
  \[
  5(2) + 6 = 10 + 6 = 16, \quad 6(2) = 12
  \]
  \[
  16 > 12 \quad \text{(True)}
  \]

- \(x = 3\):
  \[
  5(3) + 6 = 15 + 6 = 21, \quad 6(3) = 18
  \]
  \[
  21 > 18 \quad \text{(True)}
  \]

- \(x = 8\):
  \[
  5(8) + 6 = 40 + 6 = 46, \quad 6(8) = 48
  \]
  \[
  46 > 48 \quad \text{(False)}
  \]

### Conclusion:
The value \(x = 8\) makes the inequality false, providing a counterexample to the conjecture. Therefore, the correct counterexample is \(x = 8\), and the correct choice should have been \(x = 8\).

Would you like further details on this? Here are five related questions for deeper understanding:

1. How can you graph the inequality \(5x + 6 > 6x\) to visualize the solution?
2. How do counterexamples disprove conjectures in mathematics?
3. What happens if you change the inequality to \(5x + 6 \geq 6x\)?
4. How do we solve more complex inequalities involving quadratic terms?
5. What are some real-life scenarios where inequalities are used to model situations?

**Tip:** Always simplify an inequality first before testing potential counterexamples to avoid unnecessary work.

What is a counterexample that can verify that the conjecture 5x + 6 > 6x is false for any x?

Explore how to disprove the inequality 5x + 6 > 6x by finding a counterexample. Learn the algebraic steps to simplify and verify when the inequality holds, and discover which value of x acts as a counterexample. Suitable for students in Grades 8-10.

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