Solve: `(1)/((a+b+x))=(1)/(a)+(1)/(b)+(1)/(x),[xne0,xne-(a+b)].`

Options

A. `a , -b`
B. `-a , b`
C. `-a , -b`
D. `a, b`

Explanation

To solve the equation \(\frac{1}{a+b+x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}\), where \(x \neq 0\) and \(x \neq -(a+b)\), we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form:
\[ \frac{1}{a+b+x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x} \]

### Step 2: Find a common denominator
To combine the right side, we find a common denominator, which is \(abx\):
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{x} = \frac{bx + ax + ab}{abx} \]

### Step 3: Set the equation equal
Now, we can set the left side equal to the right side:
\[ \frac{1}{a+b+x} = \frac{bx + ax + ab}{abx} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ abx = (a+b+x)(bx + ax + ab) \]

### Step 5: Expand the right side
Expanding the right side:
\[ abx = (a+b+x)(bx + ax + ab) \]
This leads to:
\[ abx = (a+b)bx + (a+b)ax + (a+b)ab + x(bx + ax + ab) \]

### Step 6: Rearrange the equation
Rearranging the equation to one side gives:
\[ 0 = (a+b)bx + (a+b)ax + (a+b)ab + x(bx + ax + ab) - abx \]

### Step 7: Combine like terms
Combining like terms leads to a quadratic equation in \(x\):
\[ 0 = -x^2 - (a+b)x - ab \]
Multiplying through by -1 gives:
\[ x^2 + (a+b)x + ab = 0 \]

### Step 8: Factor the quadratic equation
We can factor the quadratic:
\[ (x + a)(x + b) = 0 \]

### Step 9: Solve for \(x\)
Setting each factor to zero gives us the solutions:
\[ x + a = 0 \quad \Rightarrow \quad x = -a \]
\[ x + b = 0 \quad \Rightarrow \quad x = -b \]

### Final Solution
Thus, the solutions to the equation are:
\[ x = -a \quad \text{and} \quad x = -b \]

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