How many positive integers are there in the solution set of `(x)/(x-2) gt 5` ?

Options

A. 0
B. 2
C. 4
D. 5

Explanation

To solve the inequality \(\frac{x}{x-2} > 5\), we will follow these steps:

### Step 1: Rewrite the Inequality
Start by rewriting the inequality:
\[ \frac{x}{x-2} > 5 \]

### Step 2: Subtract 5 from Both Sides
To eliminate the fraction, we can subtract 5 from both sides:
\[ \frac{x}{x-2} - 5 > 0 \]

### Step 3: Combine the Terms
To combine the terms on the left side, we need a common denominator:
\[ \frac{x - 5(x-2)}{x-2} > 0 \]
This simplifies to:
\[ \frac{x - 5x + 10}{x-2} > 0 \]
\[ \frac{-4x + 10}{x-2} > 0 \]

### Step 4: Factor the Numerator
Now, we can factor the numerator:
\[ \frac{-2(2x - 5)}{x-2} > 0 \]

### Step 5: Identify Critical Points
To find the critical points, set the numerator and denominator to zero:
1. \( -2(2x - 5) = 0 \) gives \( 2x - 5 = 0 \) or \( x = \frac{5}{2} = 2.5 \)
2. \( x - 2 = 0 \) gives \( x = 2 \)

### Step 6: Test Intervals
Now we test the intervals determined by the critical points \(x = 2\) and \(x = 2.5\):
- Choose a test point in the interval \((-∞, 2)\): Let \(x = 1\)
\[ \frac{-2(2(1) - 5)}{1-2} = \frac{-2(-3)}{-1} = \frac{6}{-1} < 0 \]
- Choose a test point in the interval \((2, 2.5)\): Let \(x = 2.25\)
\[ \frac{-2(2(2.25) - 5)}{2.25-2} = \frac{-2(4.5 - 5)}{0.25} = \frac{-2(-0.5)}{0.25} = \frac{1}{0.25} > 0 \]
- Choose a test point in the interval \((2.5, ∞)\): Let \(x = 3\)
\[ \frac{-2(2(3) - 5)}{3-2} = \frac{-2(6 - 5)}{1} = -2 < 0 \]

### Step 7: Determine the Solution Set
The inequality \(\frac{-2(2x - 5)}{x-2} > 0\) is satisfied in the interval \((2, 2.5)\).

### Step 8: Identify Positive Integers
Now we need to find the positive integers in the interval \((2, 2.5)\). The only positive integer in this interval is:
- 2 (but it is excluded since the expression is undefined at \(x = 2\))

### Conclusion
Thus, there are no positive integers in the solution set of the inequality \(\frac{x}{x-2} > 5\).

### Final Answer
The number of positive integers in the solution set is **0**.

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