Alright, let's solve the given rational equation step-by-step and identify the solutions and extraneous solutions.
Given equation:
[tex]\[
\frac{3}{x^2 + 5x + 6} + \frac{x-1}{x+2} = \frac{7}{x+3}
\][/tex]
### Step 1: Factor the denominators
First, we factor the quadratic denominator to simplify our work with the least common denominator (LCD).
[tex]\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\][/tex]
### Step 2: Determine the LCD
The least common denominator (LCD) of \(x^2 + 5x + 6\), \(x + 2\), and \(x + 3\) is:
[tex]\[
\text{LCD} = (x + 2)(x + 3)
\][/tex]
### Step 3: Multiply both sides by the LCD
We multiply both sides of the equation by the LCD to eliminate the denominators:
[tex]\[
(x+2)(x+3) \left(\frac{3}{(x + 2)(x + 3)} + \frac{x-1}{x + 2}\right) = (x+2)(x+3) \left(\frac{7}{x + 3}\right)
\][/tex]
### Step 4: Simplify the equation
Simplify each term:
[tex]\[
3 + (x - 1)(x + 3) = 7(x + 2)
\][/tex]
Distribute and combine like terms:
[tex]\[
3 + x^2 + 2x - 3 = 7x + 14
\][/tex]
[tex]\[
x^2 + 2x = 7x + 14
\][/tex]
### Step 5: Rearrange the equation
Move all terms to one side of the equation:
[tex]\[
x^2 + 2x - 7x - 14 = 0
\][/tex]
[tex]\[
x^2 - 5x - 14 = 0
\][/tex]
### Step 6: Solve the quadratic equation
We solve the quadratic equation by factoring, if possible, or using the quadratic formula. This factors nicely as:
[tex]\[
(x - 7)(x + 2) = 0
\][/tex]
This yields the solutions:
[tex]\[
x = 7 \quad \text{or} \quad x = -2
\][/tex]
### Step 7: Check for extraneous solutions
We substitute each solution back into the original equation to check if it is valid. Solutions that make any denominator zero are extraneous.
For \( x = 7 \):
- The denominators \( x^2 + 5x + 6 \), \( x + 2 \), and \( x + 3 \) are all non-zero.
[tex]\[
x^2 + 5x + 6 = 49 + 35 + 6 = 90 \neq 0
\][/tex]
[tex]\[
x + 2 = 7 + 2 = 9 \neq 0
\][/tex]
[tex]\[
x + 3 = 7 + 3 = 10 \neq 0
\][/tex]
So, \( x = 7 \) is a valid solution.
For \( x = -2 \):
- The denominators \( x + 2 \) is zero:
[tex]\[
x + 2 = -2 + 2 = 0
\][/tex]
This makes \( x = -2 \) an extraneous solution.
### Final Results
Thus, the solution to the equation is:
[tex]\[
x = 7 \quad \text{(valid solution)}
\][/tex]
[tex]\[
x = -2 \quad \text{(extraneous solution)}
\][/tex]
