If `(x^(2) + 1)/(x)= 3(1)/(3) and x gt 1`, find
`x- (1)/(x)`

Answer

To solve the equation \(\frac{x^2 + 1}{x} = 3 \frac{1}{3}\) and find \(x - \frac{1}{x}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{x^2 + 1}{x} = 3 \frac{1}{3} \]
We can convert the mixed fraction \(3 \frac{1}{3}\) into an improper fraction:
\[ 3 \frac{1}{3} = \frac{10}{3} \]
Thus, we rewrite the equation as:
\[ \frac{x^2 + 1}{x} = \frac{10}{3} \]

### Step 2: Clear the fraction
To eliminate the fraction, multiply both sides by \(x\):
\[ x^2 + 1 = \frac{10}{3} x \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ x^2 - \frac{10}{3} x + 1 = 0 \]

### Step 4: Multiply through by 3
To eliminate the fraction, multiply the entire equation by 3:
\[ 3x^2 - 10x + 3 = 0 \]

### Step 5: Use the quadratic formula
We can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = -10\), and \(c = 3\):
\[ b^2 - 4ac = (-10)^2 - 4 \cdot 3 \cdot 3 = 100 - 36 = 64 \]
Now substituting into the formula:
\[ x = \frac{10 \pm \sqrt{64}}{2 \cdot 3} = \frac{10 \pm 8}{6} \]

### Step 6: Calculate the values of \(x\)
Calculating the two possible values:
1. \(x = \frac{10 + 8}{6} = \frac{18}{6} = 3\)
2. \(x = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}\)

Since we are given that \(x > 1\), we take:
\[ x = 3 \]

### Step 7: Find \(x - \frac{1}{x}\)
Now we need to find \(x - \frac{1}{x}\):
\[ x - \frac{1}{x} = 3 - \frac{1}{3} = 3 - \frac{1}{3} = \frac{9}{3} - \frac{1}{3} = \frac{8}{3} \]

### Final Answer
Thus, the value of \(x - \frac{1}{x}\) is:
\[ \frac{8}{3} \]