What is a counter example that can verify that the conjecture below is false?  3z + 4 > 4z  

The conjecture is: 

\[
3z + 4 > 4z
\]

To disprove this conjecture, we need to find a counterexample where the inequality is **not** true.

### Steps:
1. Start by simplifying the inequality:
   \[
   3z + 4 > 4z
   \]
   Subtract \(3z\) from both sides:
   \[
   4 > z
   \]
   The inequality now reads \(z < 4\).

2. This suggests that any number **less than 4** satisfies the inequality, but if \(z \geq 4\), the inequality may not hold.

### Counterexample:

Let’s test \(z = 4\):

\[
3(4) + 4 > 4(4) \quad \text{or} \quad 12 + 4 > 16
\]
\[
16 > 16
\]

This is false since \(16 = 16\), not greater. Thus, \(z = 4\) is a counterexample that proves the conjecture is false.

Would you like more details on how counterexamples work, or do you have other questions?

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Here are 5 related questions:
1. What is the general method to disprove inequalities?
2. Can a conjecture be true for some values but false for others?
3. How can counterexamples be used in proving or disproving statements in algebra?
4. What is the difference between strict inequalities and non-strict ones (like \(\geq\) or \(\leq\))?
5. How can inequalities be graphically represented on a number line?

**Tip:** When looking for counterexamples, always try boundary or extreme values first, as they often provide clues!

What is a counterexample that can verify that the conjecture below is false? 3z + 4 > 4z

Explore how to disprove the inequality 3z + 4 > 4z by using a counterexample. This problem focuses on key algebraic concepts such as inequalities and provides a step-by-step solution to identify when the inequality is false. Suitable for students in Grades 6-8.

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