- Start at "START HERE!" with the expression $\frac{5}{2x} + \frac{3}{2x}$.
- Since the denominators are the same, add the numerators: $\frac{5+3}{2x} = \frac{8}{2x}$.
- Simplify the fraction: $\frac{8}{2x} = \frac{4}{x}$.
- Follow the path labeled with $\frac{4}{x}$ to the next box containing $\frac{x+5}{x+3} + \frac{x+1}{x+3}$.
- Add the numerators since the denominators are the same: $\frac{(x+5)+(x+1)}{x+3} = \frac{2x+6}{x+3}$.
- Factor the numerator: $\frac{2(x+3)}{x+3}$.
- Cancel the common factor $(x+3)$ (assuming $x \neq -3$): $2$.
- Follow the path labeled with $2$ to the next box containing $\frac{4}{x} + \frac{x+2}{2x}$.
- Find a common denominator, which is $2x$: $\frac{4 \cdot 2}{2x} + \frac{x+2}{2x} = \frac{8 + x + 2}{2x} = \frac{x+10}{2x}$.
- Follow the path labeled with $\frac{x+10}{2x}$ to the next box containing $\frac{x+5}{2x} - \frac{1}{4x}$.
- Find a common denominator, which is $4x$: $\frac{2(x+5)}{4x} - \frac{1}{4x} = \frac{2x+10-1}{4x} = \frac{2x+9}{4x}$.
- Follow the path labeled with $\frac{2x+9}{4x}$ to the next box containing $\frac{5}{x+6} + \frac{3}{x+3}$.
- Find a common denominator, which is $(x+6)(x+3)$: $\frac{5(x+3)}{(x+6)(x+3)} + \frac{3(x+6)}{(x+6)(x+3)} = \frac{5x+15+3x+18}{(x+6)(x+3)} = \frac{8x+33}{(x+6)(x+3)}$.
- Follow the path labeled with $\frac{8x+33}{(x+6)(x+3)}$ to the next box containing $\frac{12x+8}{4x-4} - \frac{10x+8}{4(x-1)}$.
- Note that $4x-4 = 4(x-1)$, so the denominators are the same: $\frac{(12x+8)-(10x+8)}{4(x-1)} = \frac{12x+8-10x-8}{4(x-1)} = \frac{2x}{4(x-1)}$.
- Simplify the fraction: $\frac{2x}{4(x-1)} = \frac{x}{2(x-1)}$.
- Follow the path labeled with $\frac{x}{2(x-1)}$ to the next box containing $\frac{2x+3}{x+2} - \frac{2x}{x+6}$.
- Find a common denominator, which is $(x+2)(x+6)$: $\frac{(2x+3)(x+6)}{(x+2)(x+6)} - \frac{2x(x+2)}{(x+2)(x+6)} = \frac{(2x^2+12x+3x+18)-(2x^2+4x)}{(x+2)(x+6)} = \frac{2x^2+15x+18-2x^2-4x}{(x+2)(x+6)} = \frac{11x+18}{(x+2)(x+6)}$.
- Follow the path labeled with $\frac{11x+18}{(x+2)(x+6)}$ to the next box containing $\frac{x-2}{x^2+2x+1} - \frac{-2x+1}{(x+1)^2}$.
- Note that $x^2+2x+1 = (x+1)^2$, so the denominators are the same: $\frac{(x-2)-(-2x+1)}{(x+1)^2} = \frac{x-2+2x-1}{(x+1)^2} = \frac{3x-3}{(x+1)^2}$.
- Factor the numerator: $\frac{3(x-1)}{(x+1)^2}$.
- Follow the path labeled with $\frac{3(x-1)}{(x+1)^2}$ to the next box containing $\frac{5x}{2} - \frac{3x}{5}$.
- Find a common denominator, which is $10$: $\frac{5x \cdot 5}{10} - \frac{3x \cdot 2}{10} = \frac{25x-6x}{10} = \frac{19x}{10}$.
- Follow the path labeled with $\frac{19x}{10}$ to the next box containing $\frac{3}{x+5} - \frac{2x}{x^2-3x-40}$.
- Factor the denominator of the second term: $x^2-3x-40 = (x+5)(x-8)$.
- The common denominator is $(x+5)(x-8)$: $\frac{3(x-8)}{(x+5)(x-8)} - \frac{2x}{(x+5)(x-8)} = \frac{3x-24-2x}{(x+5)(x-8)} = \frac{x-24}{(x+5)(x-8)}$.
- Follow the path labeled with $\frac{x-24}{(x+5)(x-8)}$ to the next box containing $\frac{5}{x} - \frac{4}{x+2}$.
- Find a common denominator, which is $x(x+2)$: $\frac{5(x+2)}{x(x+2)} - \frac{4x}{x(x+2)} = \frac{5x+10-4x}{x(x+2)} = \frac{x+10}{x(x+2)}$.
- Follow the path labeled with $\frac{x+10}{x(x+2)}$ to the next box containing $\frac{5x+6}{4(x+2)}$.
- This is the final answer.
Parent Tip: Review the logic above to help your child master the concept of adding and subtracting rational expressions worksheet pdf.
