Express each of the following as a rational expression.
`((x+3))/((x-2))-((x+1))/((x-3))` :

Options

A. `(x-7)/((x^(2)+5x-6))`
B. `(7-x)/((x^(2)-5x+6))`
C. `(x-7)/(2x^(2)-5x+6)`
D. `(x-7)/(x^(2)-5x+6)`

Answer

To express the given expression \(\frac{x+3}{x-2} - \frac{x+1}{x-3}\) as a rational expression, we will follow these steps:

### Step 1: Identify the denominators
The denominators of the two fractions are \(x - 2\) and \(x - 3\).

### Step 2: Find the Least Common Denominator (LCD)
The least common denominator (LCD) of the two fractions is the product of the two denominators:
\[ \text{LCD} = (x - 2)(x - 3) \]

### Step 3: Rewrite each fraction with the LCD
We need to rewrite each fraction so that they have the same denominator, which is the LCD.

For the first fraction \(\frac{x+3}{x-2}\):
\[ \frac{x+3}{x-2} = \frac{(x+3)(x-3)}{(x-2)(x-3)} \]

For the second fraction \(\frac{x+1}{x-3}\):
\[ \frac{x+1}{x-3} = \frac{(x+1)(x-2)}{(x-3)(x-2)} \]

### Step 4: Combine the fractions
Now we can combine the two fractions over the common denominator:
\[ \frac{(x+3)(x-3) - (x+1)(x-2)}{(x-2)(x-3)} \]

### Step 5: Simplify the numerator
Now we will simplify the numerator:
1. Expand \((x+3)(x-3)\):
\[ (x+3)(x-3) = x^2 - 9 \]

2. Expand \((x+1)(x-2)\):
\[ (x+1)(x-2) = x^2 - 2x + x - 2 = x^2 - x - 2 \]

3. Now substitute these back into the numerator:
\[ x^2 - 9 - (x^2 - x - 2) = x^2 - 9 - x^2 + x + 2 \]
This simplifies to:
\[ x - 7 \]

### Step 6: Write the final expression
Now we can write the final expression as:
\[ \frac{x - 7}{(x - 2)(x - 3)} \]

### Final Answer
Thus, the expression \(\frac{x+3}{x-2} - \frac{x+1}{x-3}\) as a rational expression is:
\[ \frac{x - 7}{(x - 2)(x - 3)} \]

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