Understanding the Truth of Mathematical Statements

The video tutorial explores whether the statement '3x is greater than or equal to x for all real numbers x' is always, sometimes, or never true. By testing with different values of x, including zero, positive, and negative numbers, it is determined that the statement is sometimes true. It holds for non-negative numbers but fails for negative numbers, as shown by a counterexample.

What is the initial statement being evaluated in the video?

• 3x is less than x for all real numbers x.

• 3x is equal to x for all real numbers x.

• 3x is greater than or equal to x for all real numbers x.

• 3x is not equal to x for all real numbers x.

When x is 0, what is the relationship between 3x and x?

• 3x is less than x.

• 3x is not comparable to x.

• 3x is equal to x.

• 3x is greater than x.

What conclusion can be drawn when x is a positive number?

• 3x is always less than x.

• 3x is never greater than or equal to x.

• 3x is sometimes less than x.

• 3x is always greater than or equal to x.

What type of number provides a counterexample to the statement?

• Zero

• Negative numbers

• Even numbers

• Positive numbers

What happens when x is -1 in the context of the statement?

• 3x is less than x.

• 3x is equal to x.

• 3x is greater than x.

• 3x is not comparable to x.

Why is the statement not 'always true'?

• Because it is false for some negative numbers.

• Because it is true for all numbers.

• Because it is true for all positive numbers.

• Because it is true for zero.

For which type of numbers is the statement '3x is greater than or equal to x' true?

• Only positive numbers

• Non-negative numbers

• Only negative numbers

• All real numbers

What is the final conclusion about the truth of the statement?

• It is never true.

• It is always true.

• It is sometimes true.

• It is true for all numbers.