Solve for `x : (1)/(a + b + x) = (1)/(a) + (1)/(b) + (1)/(x) , a ne b ne 0 , x ne 0 , x ne -(a + b)`

Answer

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

### Step 1: Rewrite the equation
Start by rewriting the equation to isolate the terms involving \(x\):
\[ \frac{1}{a + b + x} - \frac{1}{x} = \frac{1}{a} + \frac{1}{b} \]

### Step 2: Find a common denominator
The left-hand side can be simplified by finding a common denominator:
\[ \frac{x - (a + b + x)}{(a + b + x)x} = \frac{1}{a} + \frac{1}{b} \]
This simplifies to:
\[ \frac{- (a + b)}{(a + b + x)x} = \frac{1}{a} + \frac{1}{b} \]

### Step 3: Simplify the right-hand side
The right-hand side can be combined into a single fraction:
\[ \frac{b + a}{ab} = \frac{a + b}{ab} \]
Now we have:
\[ \frac{-(a + b)}{(a + b + x)x} = \frac{a + b}{ab} \]

### Step 4: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ -(a + b) \cdot ab = (a + b + x)x \]

### Step 5: Expand both sides
Expanding both sides gives:
\[ -ab(a + b) = ax + bx + x^2 \]

### Step 6: Rearrange the equation
Rearranging the equation leads to:
\[ x^2 + (a + b)x + ab(a + b) = 0 \]

### Step 7: Factor the quadratic equation
This is a quadratic equation in standard form \(Ax^2 + Bx + C = 0\). We can factor it as follows:
\[ x^2 + (a + b)x + ab(a + b) = 0 \]
Factoring out gives:
\[ (x + a)(x + b) = 0 \]

### Step 8: Solve for \(x\)
Setting each factor to zero gives the solutions:
\[ x + a = 0 \quad \Rightarrow \quad x = -a \]
\[ x + b = 0 \quad \Rightarrow \quad x = -b \]

### Final Solutions
Thus, the solutions for \(x\) are:
\[ x = -a \quad \text{or} \quad x = -b \]