The set of values of x for which `|(x - 5)x| > 0` is :

Options

A. a. all real numbers
B. b. `R - {0}`
C. c. `R - {0, 5}`
D. d. `R - {0,5}`

Answer

To solve the inequality \( |(x - 5)x| > 0 \), we need to analyze the expression inside the absolute value and determine when it is not equal to zero.

### Step 1: Set the expression inside the absolute value to zero
We start by finding when the expression \( (x - 5)x \) equals zero:

\[ (x - 5)x = 0 \]

This equation is satisfied when either factor is zero:

1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x = 0 \)

### Step 2: Determine the intervals
The critical points from Step 1 divide the number line into three intervals:

1. \( (-\infty, 0) \)
2. \( (0, 5) \)
3. \( (5, \infty) \)

### Step 3: Test each interval
We will test a point from each interval to see if \( (x - 5)x > 0 \).

1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \)
\[ (-1 - 5)(-1) = (-6)(-1) = 6 > 0 \]
So, the inequality holds.

2. **Interval \( (0, 5) \)**: Choose \( x = 1 \)
\[ (1 - 5)(1) = (-4)(1) = -4 < 0 \]
So, the inequality does not hold.

3. **Interval \( (5, \infty) \)**: Choose \( x = 6 \)
\[ (6 - 5)(6) = (1)(6) = 6 > 0 \]
So, the inequality holds.

### Step 4: Combine the results
From our tests, we find that the inequality \( |(x - 5)x| > 0 \) holds for the intervals \( (-\infty, 0) \) and \( (5, \infty) \). It does not hold at \( x = 0 \) and \( x = 5 \) since both make the expression equal to zero.

### Final Solution
The set of values of \( x \) for which \( |(x - 5)x| > 0 \) is:

\[ (-\infty, 0) \cup (5, \infty) \]

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