If `(x + 1/x) : (x-1/x) = 5:3`, then the value of x is/are

Options

A. `+-4`
B. `+-2`
C. `+-3`
D. 0

Answer

To solve the problem `(x + 1/x) : (x - 1/x) = 5 : 3`, we can follow these steps:

### Step 1: Set up the equation using the ratio
Given the ratio, we can express it as:
\[ \frac{x + \frac{1}{x}}{x - \frac{1}{x}} = \frac{5}{3} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3 \left(x + \frac{1}{x}\right) = 5 \left(x - \frac{1}{x}\right) \]

### Step 3: Distribute both sides
Distributing both sides results in:
\[ 3x + \frac{3}{x} = 5x - \frac{5}{x} \]

### Step 4: Rearrange the equation
Now, let's move all terms involving \(x\) to one side and constant terms to the other side:
\[ 3x + \frac{3}{x} + \frac{5}{x} = 5x \]
This simplifies to:
\[ 3x + \frac{8}{x} = 5x \]

### Step 5: Isolate the terms
Rearranging gives:
\[ \frac{8}{x} = 5x - 3x \]
\[ \frac{8}{x} = 2x \]

### Step 6: Multiply both sides by \(x\) to eliminate the fraction
Multiplying both sides by \(x\) (assuming \(x \neq 0\)):
\[ 8 = 2x^2 \]

### Step 7: Solve for \(x^2\)
Dividing both sides by 2:
\[ x^2 = 4 \]

### Step 8: Take the square root
Taking the square root of both sides gives:
\[ x = \pm 2 \]

### Final Answer
Thus, the values of \(x\) are:
\[ x = 2 \quad \text{or} \quad x = -2 \]

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