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Chapter 0: Problem 56

Explain why $$ |-a|=|a| $$ for all real numbers \(a\).

Short Answer

Expert verified
The absolute value function measures the distance of any number from the origin, 0, on the real number line. We can prove that \(|-a| = |a|\) for all real numbers \(a\) by considering two cases. 1. If \(a \ge 0\), we have \(-a \le 0\), so applying the absolute value definition, we obtain \(|-a| = -(-a) = a\), which equals \(|a|\). 2. If \(a < 0\), we have \(-a > 0\), so applying the absolute value definition, we obtain \(|-a| = -a\). Likewise, applying the definition to \(a\), we get \(|a| = -a\). Therefore, in this case, \(|-a| = |a|\) as well. Thus, the equation \(|-a| = |a|\) holds true for all real numbers \(a\).

Step by step solution

01

Definition of Absolute Value

Recall that the absolute value of a real number \(a\) is defined as: \[|a| = \begin{cases} a & \text{if } a \ge 0, \\ -a & \text{if } a < 0, \end{cases}\] The absolute value function measures the distance of any number from the origin, 0, on the real number line.
02

Case 1: \(a \ge 0\)

If \(a \ge 0\), we'll work through the expression \(|-a|\) to find its absolute value. Since \(a\) is a non-negative number, \(-a\) will be a non-positive number. Therefore, \(-a \le 0\), and we know that applying the absolute value definition to \(-a\) yields: \[|-a| = -(-a) = a\] Now we can compare \(|-a|\) to \(|a|\): \[|-a| = a = |a|\] In this case, \(|-a| = |a|\) holds true for all non-negative real numbers \(a\).
03

Case 2: \(a < 0\)

If \(a < 0\), we'll work through the expression \(|-a|\) to find its absolute value. Since \(a\) is a negative number, \(-a\) will be a positive number. Therefore, \(-a > 0\), and we know that applying the absolute value definition to \(-a\) yields: \[|-a| = -a\] Now we can compare \(|-a|\) to \(|a|\): \[|-a| = -a\] Applying the absolute value definition to \(a\), we'll find: \[|a| = -a\] In this case, \(|-a| = |a|\) holds true for all negative real numbers \(a\).
04

Conclusion

Therefore, we have shown that the equation \(|-a| = |a|\) holds true for all real numbers \(a\), no matter if they are positive, negative, or zero.

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