If ` ( x)/( a + b) + 1 = ( x)/( a - b) + ( a - b)/( a + b)` then x is equal to

Options

A. 2a-b
B. a+b
C. a-b
D. 2a+b

Answer

To solve the equation

\[ \frac{x}{a + b} + 1 = \frac{x}{a - b} + \frac{a - b}{a + b} \]

we will follow these steps:

### Step 1: Rearranging the equation
First, we will isolate \( x \) on one side of the equation. We can do this by subtracting \( \frac{x}{a - b} \) from both sides:

\[ \frac{x}{a + b} + 1 - \frac{x}{a - b} = \frac{a - b}{a + b} \]

### Step 2: Finding a common denominator
Next, we need to combine the fractions on the left side. The common denominator for \( (a + b) \) and \( (a - b) \) is \( (a + b)(a - b) \). We rewrite the left side:

\[ \frac{x(a - b)}{(a + b)(a - b)} + \frac{(a + b)(a - b)}{(a + b)(a - b)} - \frac{x(a + b)}{(a + b)(a - b)} = \frac{a - b}{a + b} \]

This simplifies to:

\[ \frac{x(a - b) + (a + b)(a - b) - x(a + b)}{(a + b)(a - b)} = \frac{a - b}{a + b} \]

### Step 3: Simplifying the numerator
Now we simplify the numerator:

\[ x(a - b) - x(a + b) + (a + b)(a - b) = x(a - b - a - b) + (a^2 - b^2) \]

This simplifies to:

\[ -x(2b) + (a^2 - b^2) \]

### Step 4: Setting the equation
Now we can set the equation:

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

### Step 5: Cross-multiplying
Cross-multiplying gives us:

\[ (-x(2b) + (a^2 - b^2))(a + b) = (a - b)(a - b)(a - b) \]

### Step 6: Solving for \( x \)
Now we can solve for \( x \):

\[ -x(2b)(a + b) + (a^2 - b^2)(a + b) = (a - b)^2 \]

Rearranging gives:

\[ -x(2b)(a + b) = (a - b)^2 - (a^2 - b^2)(a + b) \]

Now we can isolate \( x \):

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

### Final Step: Simplifying
After simplifying, we find:

\[ x = a - b \]

Thus, the value of \( x \) is:

\[ \boxed{a - b} \]