Solve : `(x)/(x-1)+(x-1)/(x)=2(1)/(2)`.

Answer

To solve the equation \(\frac{x}{x-1} + \frac{x-1}{x} = 2\frac{1}{2}\), we will follow these steps:

### Step 1: Rewrite the equation
First, we can rewrite \(2\frac{1}{2}\) as \(\frac{5}{2}\):
\[ \frac{x}{x-1} + \frac{x-1}{x} = \frac{5}{2} \]

### Step 2: Find a common denominator
The common denominator for the left-hand side is \(x(x-1)\). We can rewrite the equation as:
\[ \frac{x^2 + (x-1)^2}{x(x-1)} = \frac{5}{2} \]

### Step 3: Expand the numerator
Now, we expand \((x-1)^2\):
\[ (x-1)^2 = x^2 - 2x + 1 \]
Thus, the left-hand side becomes:
\[ \frac{x^2 + (x^2 - 2x + 1)}{x(x-1)} = \frac{2x^2 - 2x + 1}{x(x-1)} \]

### Step 4: Set up the equation
Now, we can set up the equation:
\[ \frac{2x^2 - 2x + 1}{x(x-1)} = \frac{5}{2} \]

### Step 5: Cross-multiply
Cross-multiplying gives us:
\[ 2(2x^2 - 2x + 1) = 5x(x-1) \]
Expanding both sides:
\[ 4x^2 - 4x + 2 = 5x^2 - 5x \]

### Step 6: Rearrange the equation
Rearranging the equation to one side gives:
\[ 4x^2 - 5x^2 - 4x + 5x + 2 = 0 \]
This simplifies to:
\[ -x^2 + x + 2 = 0 \]
or
\[ x^2 - x - 2 = 0 \]

### Step 7: Factor the quadratic
Now we can factor the quadratic:
\[ (x - 2)(x + 1) = 0 \]

### Step 8: Solve for \(x\)
Setting each factor to zero gives us:
1. \(x - 2 = 0 \Rightarrow x = 2\)
2. \(x + 1 = 0 \Rightarrow x = -1\)

### Step 9: Conclusion
The solutions to the equation are:
\[ x = 2 \quad \text{or} \quad x = -1 \]

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