Solve the equation x + 13 = (12x – 45)/(x + 11) – (18x + 33)/(x + 11)

Explanation

Step 1: Combine fractions on right side
x + 13 = [(12x – 45) – (18x + 33)]/(x + 11)
x + 13 = (12x – 45 – 18x – 33)/(x + 11)
x + 13 = (-6x – 78)/(x + 11)

Step 2: Cross multiply
(x + 13)(x + 11) = -6x – 78

Step 3: Expand left side
x² + 11x + 13x + 143 = -6x – 78
x² + 24x + 143 = -6x – 78

Step 4: Rearrange to standard form
x² + 24x + 6x + 143 + 78 = 0
x² + 30x + 221 = 0

Step 5: Factorize
(x + 13)(x + 17) = 0
x = -13 or x = -17

Step 6: Check restrictions
x ≠ -11 (would make denominator zero)
Both -13 and -17 are valid