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The question asks for an integer such that the opposite of the number is greater than the number itself. Let's break this down mathematically:

- Let Joe's number be \( x \).
- The opposite of Joe's number is \( -x \).
- The condition given is: \( -x > x \).

To solve this inequality:

1. Add \( x \) to both sides:
   \[
   0 > 2x
   \]
   
2. Divide both sides by 2:
   \[
   0 > x \quad \text{or} \quad x < 0
   \]

This means that Joe's number must be a negative integer, since only for negative numbers does the opposite become greater than the number itself.

Thus, **Joe's number must be negative**.

Would you like further clarification or more details?

Here are 5 follow-up questions that build on this concept:

1. What happens if Joe's number is zero?
2. Can Joe's number be a fraction or does it have to be an integer?
3. What if the question was asking for a positive number—how would the inequality change?
4. Why is dividing by a negative number in inequalities different from dividing by a positive number?
5. How would this situation change if the condition was that the opposite of Joe's number is less than the number itself?

**Tip:** When dealing with inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Joe is thinking of an integer. The opposite of his number is greater than the number itself. Which of the following must be true of Joe's number?

Explore how to solve integer inequalities where the opposite of a number is greater than the number itself. This problem focuses on negative integers and basic inequality concepts, suitable for Grades 6-8.

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