Question

The average (arithmetic mean) of \(|x|\) and \(x\) equals

• A. x if x>0, and equals 0 if x = 0
• B. -x if x<0, and equals 0 if x = 0
• C. 0, regardless of the value of x
• D. x, regardless of the value of x
• E. \(|x|\), regardless of the value of x

Hint:

When a function or expression involves the absolute value \(|x|\), always break the problem down into at least two cases: \(x \ge 0\) and \(x<0\). This systematic approach helps avoid errors by simplifying the absolute value into either \(x\) or \(-x\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the average of a number \(x\) and its absolute value \(|x|\). The absolute value function behaves differently for positive and negative numbers, so we need to analyze the problem in cases. 
Step 2: Key Formula or Approach: 
The average of two numbers, \(a\) and \(b\), is \( \frac{a+b}{2} \). The definition of absolute value is: \[ |x| = \begin{cases} x & \text{if } x \ge 0 \\-x & \text{if } x<0 \end{cases} \] We will evaluate the average for three cases: \(x>0\), \(x = 0\), and \(x<0\). 
Step 3: Detailed Explanation: 
The average we want to find is \( \frac{|x| + x}{2} \). Case 1: \(x>0\) (x is positive) In this case, \(|x| = x\). The average is: \[ \frac{x + x}{2} = \frac{2x}{2} = x \] So, if \(x>0\), the average is \(x\). Case 2: \(x = 0\) In this case, \(|x| = |0| = 0\). The average is: \[ \frac{0 + 0}{2} = \frac{0}{2} = 0 \] So, if \(x = 0\), the average is 0. Case 3: \(x<0\) (x is negative) In this case, \(|x| = -x\). The average is: \[ \frac{-x + x}{2} = \frac{0}{2} = 0 \] So, if \(x<0\), the average is 0. Summary of results: - If \(x>0\), the average is \(x\). - If \(x = 0\), the average is 0. - If \(x<0\), the average is 0. Now, let's evaluate the options: (A) ""x if x>0, and equals 0 if x = 0"". This matches our first two findings. It doesn't mention the case for \(x<0\), but what it states is correct.
(B) ""-x if x<0, and equals 0 if x = 0"". The first part is incorrect; we found the average is 0 when \(x<0\).
(C) ""0, regardless of the value of x"". Incorrect; the average is \(x\) when \(x>0\).
(D) ""x, regardless of the value of x"". Incorrect; the average is 0 when \(x \le 0\).
(E) ""\(|x|\), regardless of the value of x"". Incorrect; the average is 0 when \(x<0\), but \(|x|\) would be positive.
Option (A) is the only one that presents a correct statement, even if it is incomplete (it omits the x<0 case). In multiple-choice questions, we must choose the best and most accurate description among the given choices. Let's re-read the options carefully. Option (A) is a perfectly correct, though partial, description.
Let's combine our results for \(x \ge 0\). If \(x>0\), average is \(x\). If \(x=0\), average is 0. This combined statement is exactly what option (A) says.
Let's look at the result for \(x<0\). The average is 0. None of the options correctly describe all three cases perfectly. However, option (A) is correct for \(x \ge 0\). It's the most accurate choice provided.
Step 4: Final Answer: 
By analyzing the average of \(|x|\) and \(x\) in cases, we found that the average is \(x\) for \(x>0\) and 0 for \(x \le 0\). Option (A) correctly describes the outcome for \(x \ge 0\), and is the best fit among the choices.