Example: Subtracting and Simplifying
Simplify the following expression:
\[ \dfrac{x+1}{x+2} - \dfrac{x-1}{(x+2)^2} \]View Step-by-Step Solution
Step 1: Set a common denominator
The denominators are \( (x+2) \) and \( (x+2)^2 \). The least common denominator is \( (x+2)^2 \). Multiply the numerator and denominator of the first term by \( \color{red}{(x+2)} \):
\[ \dfrac{(x+1)\color{red}{(x+2)}}{(x+2)\color{red}{(x+2)}} - \dfrac{x-1}{(x+2)^2} \]Step 2: Combine into a single fraction
\[ = \dfrac{(x+1)(x+2) - (x-1)}{(x+2)^2} \]Step 3: Expand the terms in the numerator
\[ = \dfrac{(x^2+3x+2) - (x-1)}{(x+2)^2} \]Step 4: Distribute the negative sign and group like terms
\[ = \dfrac{x^2+3x+2 - x + 1}{(x+2)^2} \] \[ = \dfrac{x^2+2x+3}{(x+2)^2} \]Final Solution:
\[ \dfrac{x+1}{x+2} - \dfrac{x-1}{(x+2)^2} = \dfrac{x^2+2x+3}{(x+2)^2} \]