Solve for x : `(1)/(x + 4) - (1)/(x-7) = (11)/(30), x ne 4, 7`

Answer

To solve the equation \(\frac{1}{x + 4} - \frac{1}{x - 7} = \frac{11}{30}\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the left-hand side is \((x + 4)(x - 7)\).

### Step 2: Rewrite the equation
Using the common denominator, we can rewrite the equation as:
\[ \frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30} \]

### Step 3: Simplify the numerator
Now simplify the numerator:
\[ (x - 7) - (x + 4) = x - 7 - x - 4 = -11 \]
So the equation becomes:
\[ \frac{-11}{(x + 4)(x - 7)} = \frac{11}{30} \]

### Step 4: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ -11 \cdot 30 = 11 \cdot (x + 4)(x - 7) \]
This simplifies to:
\[ -330 = 11(x^2 - 3x - 28) \]

### Step 5: Distribute the 11
Distributing the 11 on the right-hand side gives:
\[ -330 = 11x^2 - 33x - 308 \]

### Step 6: Rearrange the equation
Rearranging the equation to set it to zero:
\[ 11x^2 - 33x - 308 + 330 = 0 \]
This simplifies to:
\[ 11x^2 - 33x + 22 = 0 \]

### Step 7: Divide the equation by 11
To simplify, divide the entire equation by 11:
\[ x^2 - 3x + 2 = 0 \]

### Step 8: Factor the quadratic equation
Now, we can factor the quadratic:
\[ (x - 1)(x - 2) = 0 \]

### Step 9: Solve for x
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 10: Verify the solutions
We need to check that neither solution is equal to 4 or 7, which they are not.

### Final Solutions:
Thus, the solutions for \(x\) are:
\[ x = 1 \quad \text{and} \quad x = 2 \]

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