Problem 52 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The statement 'The opposite of \(|a|\) is never positive' is true.

Step 1: Understand absolute value

First, it's crucial to understand the absolute value notation. For a real number a, the absolute value, denoted by \(|a|\), gives the 'magnitude' of a without consideration of its direction. Therefore, \(|a|\) is always non-negative.

Step 2: Consider the opposite

Next, consider the phrase 'the opposite of \(|a|\)'. This refers to \(-|a|\), which is the negative of the absolute value of a.

Step 3: Determine whether the statement is true

Since \(|a|\) is always non-negative, its opposite \(-|a|\) will always be non-positive. Therefore, the statement that \(-|a|\) is never positive is true.

Key Concepts

Algebraic Reasoning

Algebraic reasoning is a method of thought that involves understanding and manipulating algebraic concepts to solve problems. It encompasses recognizing patterns, making conjectures, and forming and substantiating arguments. When applied to the context of absolute values, algebraic reasoning assists us in comprehending the inherent properties of numbers and operations performed on them.

For instance, in the exercise given, algebraic reasoning is employed to evaluate the truthfulness of a statement concerning absolute values. This is done by breaking down the statement into more familiar terms, such as understanding that the absolute value of any real number is its distance from zero on a number line, which is always a non-negative number. Through algebraic reasoning, we can assert that since the absolute value of a number is never negative, the opposite of said absolute value, indicated by ega{egativethinmathspace{-}}|a| ega{egativethinmathspace{}} egnegmiddlespacebymi{egmediumspace}ega{egativethinmathspace{}} egmathpunct{egenmediumspace}egathinmathspace ega{egathinmathspace{}} egmedmathspace{}} egawidespace, must always result in a non-positive value.

Absolute Value Notation

Absolute value notation is vital in communicating the concept of magnitude without regard to sign. It is represented by two vertical bars surrounding a number or expression, like this: ega{|a|}egathinmathspace ega{egathinmathspace{}} egmathpunct{egmediumspace}ega{egativethinmathspace{}} egnegmiddlespace . This mathematical notation signifies the distance of the number 'a' from zero, regardless of its direction on the number line. Thus, whether 'a' is positive or negative, the absolute value of 'a' is never a negative number.

To elaborate, if 'a' is a positive number, its absolute value is just 'a' itself. If 'a' is negative, then its absolute value is '-a', which turns it into a positive number. For example, the absolute value of 5 is 5 (ega{|5| = 5}egmathpunct{egathinmathspace}egathinmathspace egmathstyle{egmedmathspace}egathinmathspace egmathcolor{egmedmathspace}egmathsymbol{egmediumpoint}ega{egativethinmathspace{}} ega{egathinmathspace{}} ega{egativethinmathspace{}} egamespace ega{=}egathinmathspace egmediumpoint ega{egathinmathspace{}} egawidespace, and the absolute value of -5 is also 5 (ega{|-5| = 5}ega{ negativethinmathspace }egawidespace , even though -5 is a negative number.

Counterexample Method

The counterexample method is a logical process used to disprove a statement by showing that one instance where the statement would not hold true does exist. This method plays a critical role in mathematics because it allows us to invalidate universal statements that declare something to be true for all possible cases.

In the context of our exercise, we would look for a counterexample to the assertion that 'the opposite of ega{|a|}egawidespace is never positive' if we believed the statement could be false. However, since ega{|a|}egawidespace by definition is never negative, and taking the opposite would imply multiplying by -1, it confirms that ega{-|a|}egawidespace would indeed never be positive. Therefore, in this specific circumstance, we cannot provide a counterexample because one does not exist, and hence, the original statement stands true.