If ` (5)/(x +2) =(2)/(x +1)+ 1/2and x gt 1,` what is the value of x ?

Options

A. 2
B. 3
C. 6
D. 9

Answer

To solve the equation \( \frac{5}{x + 2} = \frac{2}{x + 1} + \frac{1}{2} \) given that \( x > 1 \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ \frac{5}{x + 2} = \frac{2}{x + 1} + \frac{1}{2} \]

### Step 2: Move the term involving \( x \) to one side
Subtract \( \frac{2}{x + 1} \) from both sides:
\[ \frac{5}{x + 2} - \frac{2}{x + 1} = \frac{1}{2} \]

### Step 3: Find a common denominator
The common denominator for the left side is \( (x + 2)(x + 1) \). Rewrite the left side:
\[ \frac{5(x + 1) - 2(x + 2)}{(x + 2)(x + 1)} = \frac{1}{2} \]

### Step 4: Simplify the numerator
Distributing in the numerator:
\[ 5(x + 1) = 5x + 5 \]
\[ -2(x + 2) = -2x - 4 \]
Combining these gives:
\[ 5x + 5 - 2x - 4 = 3x + 1 \]
Thus, we have:
\[ \frac{3x + 1}{(x + 2)(x + 1)} = \frac{1}{2} \]

### Step 5: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ 2(3x + 1) = (x + 2)(x + 1) \]

### Step 6: Expand both sides
Expanding both sides:
\[ 6x + 2 = x^2 + 3x + 2 \]

### Step 7: Rearrange the equation
Move all terms to one side:
\[ 0 = x^2 + 3x + 2 - 6x - 2 \]
This simplifies to:
\[ 0 = x^2 - 3x \]

### Step 8: Factor the equation
Factoring gives:
\[ x(x - 3) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives:
\[ x = 0 \quad \text{or} \quad x = 3 \]

### Step 10: Consider the condition \( x > 1 \)
Since we are given that \( x > 1 \), we discard \( x = 0 \). Thus, the solution is:
\[ x = 3 \]

### Final Answer
The value of \( x \) is \( 3 \).

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