Problem 56 - Algebra 1

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Properties of Absolute Value

Absolute value is a way to describe how far a number is from zero on a number line, without considering its direction. It’s always a non-negative number, meaning it can be zero or positive, but never negative. For any real number \( a \), the absolute value is denoted \(|a|\). Here’s what you need to know:

• If \( a \) is positive or zero, then \(|a| = a\).
• If \( a \) is negative, then \(|a| = -a\) (which turns it into a positive number).
• Visualize \(|a|\) as the distance from zero, regardless of the direction.

With this in mind, consider the opposite of \(|a|\): it’s the same value but with the opposite sign. So, if \(|a|\) is positive or zero, its opposite will be non-positive.

Real Numbers

Real numbers include all the numbers you can think of that lie on the continuous number line. This includes:

• Integers, like -2, 0, and 3
• Fractions, like 1/2 and -3/4
• Decimals, like 0.75 and -2.5
• Irrational numbers, like \( \sqrt{2} \) and \( \pi \)

All these are considered real numbers because they have a precise position on the number line. It’s crucial to understand that the concept of absolute value applies universally to real numbers, showing how far each one is from zero without considering direction.

Positive Numbers

Positive numbers are numbers greater than zero. They do not include zero itself or any negative numbers. When discussing the opposite of an absolute value, remember that:

• The opposite of a positive number is always negative.
• The opposite of zero is still zero.

Given these principles, if \(|a|\) is a positive number, its opposite will naturally be negative, not positive. Hence, the notion that the opposite of an absolute value can't be positive holds true, aligning with general math rules.