Section 3.3
Adding Fractions
The critical step in adding or subtracting fractions is writing each fraction in an equivalent form so that all the fractions have the same common denominator. In order to facilitate factoring denominators (so that the least common denominator can be determined), make sure that all expressions are written in "standard form" with highest powers coming first and leading coefficients positive (see example d. below).
Examples:
a. \begin{align}
\frac{x}{3}+\frac{y}{4}&=\frac{x}{3}\cdot\frac{4}{4}+\frac{y}{4}\cdot\frac{3}{3}\\
&=\frac{4x}{12}+\frac{3y}{12}\\
&=\frac{4x+3y}{12}
\end{align}
b. \begin{align}
\frac{3}{x+3}+\frac{2}{x-2}&=\frac{3}{x+3}\cdot\frac{x-2}{x-2}+\frac{2}{x-2}\cdot\frac{x+3}{x+3}\\
&=\frac{3(x-2)}{(x+3)(x-2)}+\frac{2(x+3)}{(x+3)(x-2)}\\
&=\frac{(3x-6)+(2x+6)}{(x+3)(x-2)}\\
&=\frac{5x}{(x+3)(x-2)}
\end{align}
c. \begin{align}
\frac{x+3}{x^2-5x-14}-\frac{2}{x^2-4}&=\frac{x+3}{(x-7)(x+2)}-\frac{2}{(x+2)(x-2)}&&\textit{Factor first.}\\
&=\frac{x+3}{(x-7)(x+2)}\cdot \frac{x-2}{x-2}-\frac{2}{(x+2)(x-2)}\cdot \frac{x-7}{x-7}\\
&=\frac{(x+3)(x-2)}{(x-7)(x+2)(x-2)}-\frac{2(x-7)}{(x+2)(x-2)(x-7)}\\
&=\frac{(x^2+x-6)-(2x-14)}{(x-7)(x+2)(x-2)}\\
&=\frac{x^2-x+8}{(x-7)(x+2)(x-2)}
\end{align}
d. \begin{align}
\frac{x+5}{x-4}+\frac{3}{16-x^2}&=\frac{x+5}{x-4}-\frac{3}{x^2-16}&&\textit{Rewrite in standard form with}\\
&&&\textit{highest powers coming first and}\\
&&&\textit{leading coefficients positive.}\\
&=\frac{x+5}{x-4}-\frac{3}{(x-4)(x+4)}\\
&=\frac{x+5}{x-4}\cdot\frac{x+4}{x+4}-\frac{3}{(x-4)(x+4)}\\
&=\frac{(x^2+9x+20)-3}{(x-4)(x+4)}\\
&=\frac{x^2+9x+17}{(x-4)(x+4)}
\end{align}
Subtracting rational expressions: Subtracting Rational Expressions
Adding and subtracting rational expressions 3: U11_L1_T3_we3 Adding and Subtracting Rational Expressions 3
Exercises
Simplify the following expressions.
\begin{array}{l l}
1. \;\;\frac{7x}{10}-\frac{2x}{10}\;\;& 2. \;\;\frac{5x-3}{x+1}+\frac{2x}{x+1}+\frac{4}{x+1}\;\; \\
3. \;\;\frac{a-6}{a+1}-\frac{3a-4}{a+1}\;\;& 4. \;\;\frac{3x-6}{{{x}^{2}}-2x+1}+\frac{5-2x}{{{x}^{2}}-2x+1}\;\; \\
5. \;\;\frac{2}{9t}-\frac{11}{6t}\;\;& 6. \;\;\frac{3}{x-2}+\frac{3}{x+2}\;\; \\
7. \;\;\frac{x+y}{x{{y}^{2}}}+\frac{3x+y}{{{x}^{2}}y}\;\;& 8. \;\;\frac{3}{x+1}+\frac{2}{3x}\;\; \\
9. \;\;\frac{x}{x-5}+\frac{x-5}{x}\;\;& 10. \;\;\frac{7}{{{x}^{2}}+x-2}+\frac{5}{{{x}^{2}}-4x+3}\;\; \\
11. \;\;\frac{x-1}{4x}-\frac{2x+3}{x}\;\;& 12. \;\;\frac{8x}{{{x}^{2}}-16}-\frac{5}{x+4}\;\; \\
13. \;\;\frac{5x+3y}{2{{x}^{2}}y}-\frac{3x-4y}{x{{y}^{2}}}\;\;& 14. \;\;\frac{x}{x-1}+\frac{1}{1-x}\;\; \\
15. \;\;\frac{x-1}{{{x}^{2}}-6x+8}-\frac{2}{2+x-{{x}^{2}}}\;\;& 16. \;\;\frac{3}{x+1}+\frac{2}{x-1}-\frac{4}{{{x}^{2}}-1}\;\; \\
17. \;\;\frac{2}{{{x}^{2}}+2x-3}+\frac{3}{{{x}^{2}}-2x+1}\;\;& 18. \;\;\frac{2y}{{{y}^{2}}-9y+14}-\frac{y-1}{{{y}^{2}}-8y+7}\;\; \\
19. \;\;x+\frac{2}{x}\;\;& 20. \;\;\frac{p}{p+2}+\frac{p+1}{p+3}+\frac{2}{{{p}^{2}}+5p+6}\;\; \\
21. \;\;\frac{1}{{{x}^{2}}{{y}^{3}}}-\frac{3}{{{x}^{4}}y}\;\;& 22. \;\;\frac{x}{{{x}^{2}}-6x+8}-\frac{2}{2+x-{{x}^{2}}}\;\; \\
23. \;\;\frac{2}{x-2}+\frac{3}{x+2}-\frac{5}{4-{{x}^{2}}}\;\;& 24. \;\;\frac{4x}{{{x}^{2}}+2x-3}+\frac{1}{1-x}-\frac{2}{x+3}\;\; \\
25. \;\;\frac{x}{{{x}^{2}}-2x-3}-\frac{x}{{{x}^{2}}-9}-\frac{1}{{{x}^{2}}+4x+3}\;\;& 26. \;\;\frac{1}{{{x}^{2}}-x}-\frac{1}{{{x}^{2}}-1}+\frac{1}{{{x}^{2}}+x}\;\; \\
27. \;\;\frac{x-1}{x+3}+\frac{2x+1}{x-2}\;\;& 28. \;\;\frac{x-1}{4{{x}^{2}}-9}-\frac{x}{2{{x}^{2}}-x-3}+\frac{2x-3}{2{{x}^{2}}+5x+3}\;\; \\
29. \;\;\frac{x+2}{x{{(x-1)}^{2}}}-\frac{x+3}{{{x}^{2}}(x-1)}-\frac{3}{{{x}^{4}}-2{{x}^{3}}+{{x}^{2}}}\;\; & 30. \;\;\frac{1}{{{a}^{3}}(a-2)}-\frac{1}{{{a}^{2}}{{(a-2)}^{2}}}+\frac{a-2}{{{a}^{2}}{{(a-2)}^{2}}}\;\; \\
31. \;\;\frac{1}{a(a-b)}+\frac{1}{b(b-a)}+\frac{1}{4ab}\;\;& 32. \;\;\frac{1}{2a}-\frac{1}{2b}+\frac{1}{a-b}-\frac{1}{a+b}-\frac{b-a}{2ab}\;\; \\
33. \;\;3+\frac{x-y}{y}+\frac{x}{x-y}+\frac{y}{y-x}+\frac{1}{x}\;\;& 34. \;\;\frac{1}{-{{a}^{2}}+4a-3}+\frac{2}{{{a}^{2}}-9}-\frac{3}{9-6a+{{a}^{2}}}\;\; \\
35. \;\;1+\frac{x}{x+3}-\frac{2x}{x+1}\;\;& 36. \;\;\frac{x+1}{{{x}^{2}}-7x+6}+\frac{x+2}{{{x}^{2}}-4x+3}-\frac{2x+3}{{{x}^{2}}-9x+18}\;\; \\
37. \;\;\frac{1}{x-1}-\frac{1}{x+1}\;\;& 38. \;\;\frac{1}{x-1}+1\;\; \\
39. \;\;\frac{1}{x-1}-1\;\;& 40. \;\;\frac{1}{{{x}^{2}}-2x+1}-\frac{1}{1-{{x}^{2}}}\;\; \\
41. \;\;\frac{2}{x-3}-\frac{1}{x-2}-\frac{1}{x-1}\;\;& 42. \;\;\frac{1}{{{x}^{2}}-5x+6}-\frac{3}{{{x}^{2}}-7x+10}+\frac{5}{{{x}^{2}}-8x+15}\;\; \\
43. \;\;\frac{1}{x+3}-\frac{2}{x+2}+\frac{1}{x+1}\;\;& 44. \;\;\frac{3}{{{x}^{2}}-x-2}-\frac{2}{{{x}^{2}}+4x+3}-\frac{4}{{{x}^{2}}+x-6}\;\; \\
45. \;\;\frac{2}{x-1}-\frac{1}{x-3}\;\;& 46. \;\;\frac{x}{x+1}-\frac{x}{x-1}-\frac{x-3}{{{x}^{2}}-1}\;\; \\
47. \;\;\frac{2x}{3+x}+\frac{1+2x}{2-x}+\frac{25}{{{x}^{2}}+x-6}\;\;& 48. \;\;\frac{x-2}{x+1}-\frac{x+1}{1-x}-\frac{{{x}^{2}}-3x+6}{{{x}^{2}}-1}\;\; \\
49. \;\;\frac{x+1}{x}-\frac{x+2}{x-1}-\frac{3x}{x-{{x}^{2}}}\;\;& 50. \;\;\frac{2x}{3+x}+\frac{1+2x}{2-x}+\frac{25}{{{x}^{2}}+x-6}\;\; \\
51. \;\;\frac{9}{3x-6}-\frac{2}{2-x}\;\;& 52. \;\;\frac{4x-13}{{{x}^{2}}-3x-10}-\frac{3}{x+2}\;\; \\
53. \;\;\frac{7a}{{{a}^{2}}+2a-3}+\frac{2a-5}{{{a}^{2}}+5a+6}\;\;& 54. \;\;\frac{4}{{{x}^{2}}-1}+\frac{2}{x+1}-\frac{3}{1-x}\;\; \\
55. \;\;\frac{1}{{{x}^{2}}+2x-3}+\frac{2}{{{x}^{2}}-9}-\frac{1}{{{x}^{2}}-4x+3}\;\;& 56. \;\;\frac{5}{(x-1)(x-2)}+\frac{2}{(x-1)(x-3)}-\frac{1}{(x-2)(x-3)}\;\; \\
57. \;\;1-\frac{1}{x+1}+\frac{1}{x-1}\;\;& 58. \;\;\frac{1}{{{x}^{2}}-3x+2}+\frac{2}{{{x}^{2}}-6x+8}-\frac{2}{{{x}^{2}}-5x+4}\;\; \\
59. \;\;1-\frac{a}{x+a}+\frac{a}{x-a}\;\;& 60. \;\;1+\frac{a}{x+a}+\frac{2a}{x-a}\;\; \\
61. \;\;\frac{1}{x-1}-\frac{2}{x-2}+\frac{1}{x-3}\;\;& 62. \;\;\frac{1}{x-r}-\frac{1}{x-(r+2)}\;\; \\
63. \;\;1-\frac{3}{x-3}+\frac{2}{x-5}\;\;& 64. \;\;\frac{1}{x-1}-\frac{2}{x+1}+\frac{3}{1-x}-\frac{4}{-1-x}\;\; \\
65. \;\;\frac{2}{x-1}-\frac{3}{x+1}+\frac{x-3}{1-x}\;\;& 66. \;\;\frac{a}{a-1}-\frac{2a}{a+1}+\frac{3a}{1-a}\;\;
\end{array}