Find the value of `(x + a)/( x - a) + ( x + b)/( x - b) `, if ` x = (2ab)/( a + b)`

Answer

To solve the expression \(\frac{x + a}{x - a} + \frac{x + b}{x - b}\) given that \(x = \frac{2ab}{a + b}\), we will follow these steps:

### Step 1: Substitute the value of \(x\)
We start by substituting the given value of \(x\) into the expression.

\[ x = \frac{2ab}{a + b} \]

### Step 2: Rewrite the first term \(\frac{x + a}{x - a}\)
We can rewrite the first term using the value of \(x\):

\[ \frac{x + a}{x - a} = \frac{\frac{2ab}{a + b} + a}{\frac{2ab}{a + b} - a} \]

### Step 3: Simplify the numerator and denominator
Now, we simplify the numerator and denominator:

**Numerator:**
\[ \frac{2ab}{a + b} + a = \frac{2ab + a(a + b)}{a + b} = \frac{2ab + a^2 + ab}{a + b} = \frac{a^2 + 3ab}{a + b} \]

**Denominator:**
\[ \frac{2ab}{a + b} - a = \frac{2ab - a(a + b)}{a + b} = \frac{2ab - a^2 - ab}{a + b} = \frac{ab - a^2}{a + b} = \frac{a(b - a)}{a + b} \]

Thus, we have:

\[ \frac{x + a}{x - a} = \frac{\frac{a^2 + 3ab}{a + b}}{\frac{a(b - a)}{a + b}} = \frac{a^2 + 3ab}{a(b - a)} \]

### Step 4: Rewrite the second term \(\frac{x + b}{x - b}\)
Similarly, we rewrite the second term:

\[ \frac{x + b}{x - b} = \frac{\frac{2ab}{a + b} + b}{\frac{2ab}{a + b} - b} \]

**Numerator:**
\[ \frac{2ab}{a + b} + b = \frac{2ab + b(a + b)}{a + b} = \frac{2ab + ab + b^2}{a + b} = \frac{3ab + b^2}{a + b} \]

**Denominator:**
\[ \frac{2ab}{a + b} - b = \frac{2ab - b(a + b)}{a + b} = \frac{2ab - ab - b^2}{a + b} = \frac{ab - b^2}{a + b} = \frac{b(a - b)}{a + b} \]

Thus, we have:

\[ \frac{x + b}{x - b} = \frac{\frac{3ab + b^2}{a + b}}{\frac{b(a - b)}{a + b}} = \frac{3ab + b^2}{b(a - b)} \]

### Step 5: Combine the two fractions
Now we can combine both fractions:

\[ \frac{x + a}{x - a} + \frac{x + b}{x - b} = \frac{a^2 + 3ab}{a(b - a)} + \frac{3ab + b^2}{b(a - b)} \]

### Step 6: Find a common denominator and simplify
The common denominator is \(ab(a - b)\):

\[ = \frac{(a^2 + 3ab)b + (3ab + b^2)a}{ab(a - b)} \]

### Step 7: Expand and simplify the numerator
Expanding the numerator:

\[ = \frac{ab^2 + 3ab^2 + 3a^2b + ab^2}{ab(a - b)} = \frac{4ab^2 + 3a^2b}{ab(a - b)} \]

### Step 8: Factor out common terms
Factoring out \(ab\):

\[ = \frac{ab(4b + 3a)}{ab(a - b)} = \frac{4b + 3a}{a - b} \]

### Final Result
Thus, the value of the expression is:

\[ \frac{4b + 3a}{a - b} \]