Section 3.7
Miscellaneous Fraction Exercises
Simplify each of the following.
\begin{array}{l l}
1. \;\;\frac{1}{a}+\frac{2}{b}-\frac{3}{c}\;\;& 2. \;\;\frac{2}{{{x}^{2}}+9x+20}-\frac{1}{{{x}^{2}}-16}\;\; \\
3. \;\;\frac{1}{x-2}-\frac{1}{x+3}\;\;& 4. \;\;\frac{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}-4 \right)\left( {{x}^{2}}-9 \right)}{\left( {{x}^{2}}+3x+2 \right)\left( {{x}^{2}}+2x-3 \right)\left( {{x}^{2}}-5x+6 \right)}\;\; \\
5. \;\;\frac{{{x}^{2}}-6x-16}{x(2x-1)+2(2x-1)}\;\;& 6. \;\;\frac{{{x}^{2}}-3x-4}{{{x}^{2}}-8x+12}\cdot \frac{{{x}^{3}}-36x}{{{x}^{2}}+7x+6}\;\; \\
7. \;\;\frac{{{x}^{2}}-4}{12}\cdot \frac{3}{2-x}\;\;& 8. \;\;\frac{1}{{{x}^{2}}-1}+\frac{1}{1-x}\;\; \\
9. \;\;\left( 1-\frac{4}{y}+\frac{4}{{{y}^{2}}} \right)\cdot \left( 1-\frac{1}{y} \right)\div \left( 1-\frac{3}{y}+\frac{2}{{{y}^{2}}} \right)\;\;& 10. \;\;\frac{{{x}^{2}}-3x-4}{{{x}^{3}}-x}\;\; \\
11. \;\;\frac{2{{x}^{2}}-4x+2}{6{{x}^{3}}-6x}\div \frac{{{x}^{2}}+8x-9}{{{x}^{4}}+{{x}^{3}}}\;\;& 12. \;\;\frac{2x(3x+y)+5(3x+y)}{9{{x}^{2}}-{{y}^{2}}}\div \frac{6{{x}^{2}}+15x}{y-3x}\;\; \\
13. \;\;\frac{7}{3ab{{c}^{2}}}-\frac{6}{3a{{b}^{2}}c}+\frac{3}{4{{a}^{2}}bc}\;\;& 14. \;\;\frac{2}{a-1}-\frac{3}{a+1}-\frac{4a}{{{a}^{2}}-1}\;\; \\
15. \;\;x+\frac{4}{x-4}\;\;& 16. \;\;\frac{x}{{{x}^{2}}-9}+\frac{3}{9-{{x}^{2}}}\;\; \\
17. \;\;\frac{a}{a-1}-\frac{1}{a}\;\;& 18. \;\;\frac{x}{{{x}^{2}}-x-6}+\frac{2}{{{x}^{2}}+3x+2}-\frac{3}{{{x}^{2}}-2x-3}\;\; \\
19. \;\;\frac{3}{x-3}-\frac{2}{x+1}\;\;& 20. \;\;\frac{2}{{{x}^{2}}-4x-5}+\frac{3}{{{x}^{2}}-1}\;\; \\
21. \;\;\frac{\tfrac{1}{x}-\tfrac{1}{3}}{\tfrac{1}{6}+\tfrac{2}{x}}\;\;& 22. \;\;\frac{\tfrac{1}{a+1}-\tfrac{1}{a-1}}{\tfrac{1}{a+1}+\tfrac{1}{a-1}}\;\;
\end{array}
23. The Hopemobile could travel $\dfrac{12{{x}^{2}}+24x-36}{{{x}^{2}}+6x+9}$ miles in $\dfrac{3{{x}^{2}}-x-2}{x(3x+2)+3(3x+2)}$ hours. What was the speed of the Hopemobile? [Remember: (rate)×(time) = distance.]
24. Determine the quotient and remainder for each of the following long division problems.
a. $\left( {{x}^{3}}-5{{x}^{2}}+2x+8 \right)\div (x-2)$ b. $\left( {{x}^{4}}-3{{x}^{2}}-2x-8 \right)\div (x+3)$
25. Solve each of the following equations for x.
a. $\dfrac{8}{x+2}-\dfrac{8}{x-2}=\dfrac{x}{{{x}^{2}}-4}$ b. $\dfrac{8}{x-1}-\dfrac{8}{x+3}=1$