Question

Decide whether the statement is true or false. If it is false, give a counter example. The opposite of \(|a|\) is never positive.

Short answer

The statement 'The opposite of \(|a|\) is never positive' is True. There is no counter example as there does not exist any real number \(a\) such that the opposite of \(|a|\) is positive.

Step-by-step solution

Understand the Properties of Absolute Value and Negatives

The absolute value of any real number \(a\), denoted \(|a|\), is always non-negative. This means \(|a|\) is either positive or zero. The opposite (negative) of a non-negative number will either be negative or remain zero.

Examine the Possible Real Number Values

Since the opposite (negative) of a non-negative number is either negative or zero, the statement 'The opposite of \(|a|\) is never positive' is always true for all values of \(a\). For example, take any real number like \(2\), its absolute value is \(2\) but, the opposite of \(2\) is \(-2\) which is not positive.

Conclusion

Ultimately, there is no real number value for \(a\) such that the opposite of \(|a|\) is positive. So, the statement 'The opposite of \(|a|\) is never positive' is always true.