Problem 98 - College Algebra and Calculus: An Applied Approach

Question

\(\quad\) Can it ever be true that \(|a|=-a\) for a real number \(a\) ? Explain.

Step-by-Step Solution

Answer

The equation \(|a|=-a\) only holds true for the real number \(a=0\).

Step 1: Consider the properties of absolute values

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

Step 2: Analyze the equation \(|a|=-a\)

The expression \(-a\) is negative if \(a\) is positive and \(-a\) is positive if \(a\) is negative. For the equation \(|a|=-a\) to hold true, the absolute value of \(a\) should be a negative number.

Step 3: Identify possible values of \(a\)

From the properties of absolute values and the configuration of the equation \(|a|=-a\), we can identify that there is no positive real number for which \(|a|=-a\). However, for \(a=0\), \(|a|\) becomes 0 and \(-a\) also becomes 0, which makes \(|a|=-a\).

Step 4: Conclusion

Considering the characteristics of the absolute value and the conditions of the given equation, it turns out that only the real number \(a=0\) can satisfy the equation \(|a|=-a\). Therefore, the statement is only and only true for the case when \(a=0\).

Key Concepts

Understanding Real Numbers

Real numbers are a broad and inclusive classification of numbers that include all possible numbers that can be located on an infinite number line. This set of numbers includes both rational and irrational numbers.

• Rational numbers: These can be expressed as a fraction of two integers, where the denominator is not zero. For example, 1/2, 4, and -3 are all rational numbers.
• Irrational numbers: These cannot be expressed as a simple fraction. Famous examples include the square root of 2 and the value of pi (π).

Real numbers also encompass other subsets such as whole numbers, integers, and natural numbers, each with its own unique properties. When dealing with real numbers, it is crucial to understand that they can be positive, negative, or zero. This wide range of values makes it possible to solve various mathematical equations and analyze mathematical concepts like absolute values effectively.

Solving Equations with Absolute Value

Equations involving absolute values can be somewhat tricky because you must consider the non-negative property of absolute values. The absolute value of a number represents its distance from zero on the number line, which is always non-negative, meaning it can never be negative. This concept becomes particularly important when solving equations like \(|a| = -a\). To solve this kind of equation, you need to understand how the absolute value interacts with the other elements in the equation. Here are key steps when solving such equations:

• Recognize that \(|a|\), the absolute value, will always result in a zero or positive outcome.
• Determine the other side of the equation, involving negative signs, which might affect the balance needed for the solution.

For equations where \(|a|\) is equated to a potentially negative value, like \(-a\), it quickly becomes apparent that only a specific value of real numbers, in this case, zero, can satisfy the equation under standard properties of real numbers.

Properties of Absolute Values

Understanding the properties of absolute values is essential when working with equations and real numbers. The fundamental property of absolute value is that it measures the distance of a number from zero, meaning it outputs only non-negative results.Here are some key properties of absolute values that are often useful:

• For any real number \( a \), the absolute value \(|a|\) is always non-negative. This means \( |a| \geq 0 \).
• If \( a \) is greater than or equal to zero, then \(|a| = a\). Conversely, if \( a \) is less than zero, \(|a| = -a\).
• Absolute value is an even function, meaning that \(|a| = |-a|\).

These properties ensure that the expression \(|a|\) has specific predictable outcomes. Applying these characteristics, one concludes that for \(|a| = -a\) to be true, \(a\) must be only equal to zero, since zero is the only number with a non-negative absolute value that's equal to its negative value.