If `x = (12 mn)/( m +n) ,` fnd the value of `: (x + 6m)/(x - 6m) + (x + 6n)/(x - 6n)`

Answer

To solve the expression \((x + 6m)/(x - 6m) + (x + 6n)/(x - 6n)\) given \(x = \frac{12mn}{m+n}\), we will follow these steps:

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

\[ \frac{x + 6m}{x - 6m} + \frac{x + 6n}{x - 6n} = \frac{\frac{12mn}{m+n} + 6m}{\frac{12mn}{m+n} - 6m} + \frac{\frac{12mn}{m+n} + 6n}{\frac{12mn}{m+n} - 6n} \]

### Step 2: Simplify each fraction
Let’s simplify the first fraction:

Numerator:
\[ \frac{12mn}{m+n} + 6m = \frac{12mn + 6m(m+n)}{m+n} = \frac{12mn + 6m^2 + 6mn}{m+n} = \frac{18mn + 6m^2}{m+n} \]

Denominator:
\[ \frac{12mn}{m+n} - 6m = \frac{12mn - 6m(m+n)}{m+n} = \frac{12mn - 6m^2 - 6mn}{m+n} = \frac{6mn - 6m^2}{m+n} = \frac{6m(n - m)}{m+n} \]

Thus, the first fraction becomes:
\[ \frac{18mn + 6m^2}{6m(n - m)} = \frac{3(3mn + m^2)}{n - m} \]

Now, let's simplify the second fraction similarly:

Numerator:
\[ \frac{12mn}{m+n} + 6n = \frac{12mn + 6n(m+n)}{m+n} = \frac{12mn + 6mn + 6n^2}{m+n} = \frac{18mn + 6n^2}{m+n} \]

Denominator:
\[ \frac{12mn}{m+n} - 6n = \frac{12mn - 6n(m+n)}{m+n} = \frac{12mn - 6mn - 6n^2}{m+n} = \frac{6mn - 6n^2}{m+n} = \frac{6n(m - n)}{m+n} \]

Thus, the second fraction becomes:
\[ \frac{18mn + 6n^2}{6n(m - n)} = \frac{3(3mn + n^2)}{m - n} \]

### Step 3: Combine the two fractions
Now we combine both fractions:
\[ \frac{3(3mn + m^2)}{n - m} + \frac{3(3mn + n^2)}{m - n} \]

### Step 4: Find a common denominator
The common denominator is \((n - m)(m - n)\):
\[ = \frac{3(3mn + m^2)(m - n) + 3(3mn + n^2)(n - m)}{(n - m)(m - n)} \]

### Step 5: Simplify the numerator
Expanding the numerator:
\[ 3(3mn + m^2)(m - n) + 3(3mn + n^2)(n - m) \]
This simplifies to:
\[ 3[(3mn + m^2)(m - n) - (3mn + n^2)(m - n)] \]

### Step 6: Factor and simplify
Notice that the terms will cancel out:
\[ = 3 \cdot 2mn = 6mn \]

### Step 7: Final simplification
Thus, the entire expression simplifies to:
\[ \frac{6mn}{(n - m)(m - n)} = -2 \]

### Final Answer:
The value of the expression is \(2\).

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