Question

Which disjunction is always true for any real number x?

A number, x, is less than 0 or greater than 1.
A number, x, is less than 0 or greater than 0.
A number, x, is greater than 5 or less than 10.
A number, x, is greater than 5 or less than 1.

Answer 1

To determine which disjunction is always true for any real number \( x \), we need to analyze each of the given statements:

1. **A number, \( x \), is less than 0 or greater than 1.**
- This statement means \( x < 0 \) or \( x > 1 \).
- There are values of \( x \) (for example, \( x = 0.5 \)) where neither condition is true, so this statement is not always true.

2. **A number, \( x \), is less than 0 or greater than 0.**
- This statement translates to \( x < 0 \) or \( x > 0 \).
- The only number that does not satisfy either inequality is \( x = 0 \), making this disjunction false for \( x = 0 \), so it is not always true.

3. **A number, \( x \), is greater than 5 or less than 10.**
- This translates to \( x > 5 \) or \( x < 10 \).
- Since any real number \( x \) that is less than 5 will still be less than 10, this means that the disjunction is true for any \( x \), making it always true.

4. **A number, \( x \), is greater than 5 or less than 1.**
- This means \( x > 5 \) or \( x < 1 \).
- However, there are many values of \( x \) (such as \( x = 3 \)) for which this is not true, so it is not always true.

Based on this analysis, the only disjunction that is always true for any real number \( x \) is:

**A number, \( x \), is greater than 5 or less than 10.**