Let's analyze the given equation:

\[
\frac{A}{B} - \frac{B}{A} = \frac{A + B}{AB}
\]

### Step 1: Clearing the Fractions
First, multiply both sides of the equation by \(AB\) to eliminate the denominators:

\[
AB \left(\frac{A}{B} - \frac{B}{A}\right) = AB \left(\frac{A + B}{AB}\right)
\]

Simplifying, we get:

\[
A^2 - B^2 = A + B
\]

### Step 2: Rearrange the Equation
Rewriting \(A^2 - B^2 = A + B\):

\[
A^2 - B^2 - A - B = 0
\]

Factoring by grouping, we have:

\[
(A^2 - A) - (B^2 + B) = 0
\]

Or equivalently:

\[
A(A - 1) = B^2 + B
\]

### Step 3: Analyzing Possible Solutions
We seek integer solutions. This kind of equation often requires trial and error or checking feasible ranges of values for \(A\) and \(B\).

Let's consider specific cases:

1. **For \(A = 1\):**
   \[
   1(1 - 1) = B^2 + B \implies 0 = B^2 + B
   \]
   Solving \(B^2 + B = 0\), we get \(B(B + 1) = 0\), so \(B = 0\) or \(B = -1\).

2. **For \(A = -1\):**
   \[
   (-1)(-1 - 1) = B^2 + B \implies 2 = B^2 + B
   \]
   Solving \(B^2 + B - 2 = 0\), the solutions are:
   \[
   B = 1 \quad \text{or} \quad B = -2
   \]

Thus, the integer pairs \((A, B)\) that satisfy the original equation are:

- \((1, 0)\)
- \((1, -1)\)
- \((-1, 1)\)
- \((-1, -2)\)

### Conclusion
There are **four** solutions to the equation.

Would you like further details or explanations on any of the steps?

### Related Questions
1. What happens if \(A\) and \(B\) are restricted to positive integers?
2. How can we verify each solution by substituting back into the original equation?
3. What other methods could we use to solve this type of equation?
4. What happens if we change the sign in the original equation?
5. Are there non-integer solutions to the equation?

### Tip
Always check the feasibility of factored forms; they often simplify complex equations significantly.

how many solutions for A/B - B/A = (A + B)/AB

Explore the step-by-step solution to the fractional equation A/B - B/A = (A + B)/AB. Learn algebraic techniques including clearing fractions, factoring, and finding integer solutions. Suitable for high school students in Grades 9-12.

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