If `(2x+3)/((x+1)(x+2))=(A)/(x+1)+(B)/(x+2)` for all real values of `x(ne -1,-2)` , what is the value of (A+B)?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation

\[ \frac{2x + 3}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2} \]

for constants \(A\) and \(B\), we can follow these steps:

### Step 1: Combine the Right-Hand Side
Start by combining the fractions on the right-hand side:

\[ \frac{A}{x + 1} + \frac{B}{x + 2} = \frac{A(x + 2) + B(x + 1)}{(x + 1)(x + 2)} \]

### Step 2: Expand the Numerator
Now, expand the numerator:

\[ A(x + 2) + B(x + 1) = Ax + 2A + Bx + B = (A + B)x + (2A + B) \]

### Step 3: Set the Numerators Equal
Since the denominators are the same, we can set the numerators equal to each other:

\[ 2x + 3 = (A + B)x + (2A + B) \]

### Step 4: Compare Coefficients
Now, compare the coefficients of \(x\) and the constant terms from both sides:

1. From the coefficients of \(x\):
\[ A + B = 2 \quad \text{(Equation 1)} \]

2. From the constant terms:
\[ 2A + B = 3 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have a system of linear equations:

1. \(A + B = 2\)
2. \(2A + B = 3\)

We can solve these equations.

From Equation 1, we can express \(B\) in terms of \(A\):

\[ B = 2 - A \]

Substituting \(B\) into Equation 2:

\[ 2A + (2 - A) = 3 \]

This simplifies to:

\[ 2A + 2 - A = 3 \]

\[ A + 2 = 3 \]

\[ A = 1 \]

### Step 6: Find \(B\)
Now substitute \(A = 1\) back into Equation 1 to find \(B\):

\[ 1 + B = 2 \implies B = 1 \]

### Step 7: Calculate \(A + B\)
Finally, we find \(A + B\):

\[ A + B = 1 + 1 = 2 \]

Thus, the value of \(A + B\) is

\[ \boxed{2} \]