`(1)/((1)/(x-4)-(1)/(x+6))`
Which of the following is equivalent to the expression above?

Options

A. `10`
B. `(10(x-4))/(x+6)`
C. `(x^(2)+2x-24)/(10)`
D. `(10)/(x^(2)+2x-24)`

Explanation

To solve the expression \((1)/((1)/(x-4)-(1)/(x+6))\), we will follow these steps:

### Step 1: Rewrite the Denominator
We start by rewriting the denominator:
\[ D = \frac{1}{x-4} - \frac{1}{x+6} \]

### Step 2: Find a Common Denominator
The denominators \(x-4\) and \(x+6\) do not have a common denominator, so we need to find the least common multiple (LCM) of these two denominators. The LCM is \((x-4)(x+6)\).

### Step 3: Rewrite Each Fraction
We will rewrite each fraction with the common denominator:
\[ D = \frac{(x+6) - (x-4)}{(x-4)(x+6)} \]

### Step 4: Simplify the Numerator
Now, simplify the numerator:
\[ D = \frac{x + 6 - x + 4}{(x-4)(x+6)} = \frac{10}{(x-4)(x+6)} \]

### Step 5: Substitute Back into the Original Expression
Now we substitute \(D\) back into the original expression:
\[ \frac{1}{D} = \frac{1}{\frac{10}{(x-4)(x+6)}} \]

### Step 6: Simplify the Expression
When we divide by a fraction, we multiply by its reciprocal:
\[ \frac{1}{D} = \frac{(x-4)(x+6)}{10} \]

### Final Answer
Thus, the expression \((1)/((1)/(x-4)-(1)/(x+6))\) simplifies to:
\[ \frac{(x-4)(x+6)}{10} \]