2.EC: Exercises for Rational Expressions and Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
Simplifying, Multiplying, and Dividing Rational Expressions
In the following exercises, determine the values for which the rational expression is undefined.
1. \dfrac{5 a+3}{3 a-2}
- Answer
-
a \neq \dfrac{2}{3}
2. \dfrac{b-7}{b^{2}-25}
3. \dfrac{5 x^{2} y^{2}}{8 y}
- Answer
-
y \neq 0
4. \dfrac{x-3}{x^{2}-x-30}
In the following exercises, simplify.
5. \dfrac{9 m^{4}}{18 m n^{3}}
6. \dfrac{x^{2}+7 x+12}{x^{2}+8 x+16}
- Answer
-
\dfrac{x+3}{x+4}
7. \dfrac{7 v-35}{25-v^{2}}
In the following exercises, perform the indicated operations and simplify results.
8. \dfrac{3 x y^{2}}{8 y^{3}} \cdot \dfrac{16 y^{2}}{24 x}
9. \dfrac{72 x-12 x^{2}}{8 x+32} \cdot \dfrac{x^{2}+10 x+24}{x^{2}-36}
- Answer
-
\dfrac{-3 x}{2}
10. \dfrac{6 y^{2}-2 y-10}{9-y^{2}} \cdot \dfrac{y^{2}-6 y+9}{6 y^{2}+29 y-20}
11. \dfrac{x^{2}-4 x-12}{x^{2}+8 x+12} \div \dfrac{x^{2}-36}{3 x}
- Answer
-
\dfrac{3 x}{(x+6)(x+6)}
12. \dfrac{y^{2}-16}{4} \div \dfrac{y^{3}-64}{2 y^{2}+8 y+32}
13. \dfrac{11+w}{w-9} \div \dfrac{121-w^{2}}{9-w}
- Answer
-
\dfrac{1}{11+w}
14. \dfrac{3 y^{2}-12 y-63}{4 y+3} \div\left(6 y^{2}-42 y\right)
15. \dfrac{\dfrac{c^{2}-64}{3 c^{2}+26 c+16}}{\dfrac{c^{2}-4 c-32}{15 c+10}}
- Answer
-
\dfrac{5}{c+4}
16. \dfrac{8 a^{2}+16 a}{a-4} \cdot \dfrac{a^{2}+2 a-24}{a^{2}+7 a+10} \div \dfrac{2 a^{2}-6 a}{a+5}
Adding and Subtracting Rational Expressions
In the following exercises, perform the indicated operations and simplify results.
1. \dfrac{4 a^{2}}{2 a-1}-\dfrac{1}{2 a-1}
2. \dfrac{y^{2}+10 y}{y+5}+\dfrac{25}{y+5}
- Answer
-
y+5
3. \dfrac{7 x^{2}}{x^{2}-9}+\dfrac{21 x}{x^{2}-9}
4. \dfrac{x^{2}}{x-7}-\dfrac{3 x+28}{x-7}
- Answer
-
x+4
5. \dfrac{y^{2}}{y+11}-\dfrac{121}{y+11}
6. \dfrac{4 q^{2}-q+3}{q^{2}+6 q+5}-\dfrac{3 q^{2}-q-6}{q^{2}+6 q+5}
- Answer
-
\dfrac{q-3}{q+5}
7. \dfrac{5 t+4 t+3}{t^{2}-25}-\dfrac{4 t^{2}-8 t-32}{t^{2}-25}
8. \dfrac{18 w}{6 w-1}+\dfrac{3 w-2}{1-6 w}
- Answer
-
\dfrac{15 w+2}{6 w-1}
9. \dfrac{a^{2}+3 a}{a^{2}-4}-\dfrac{3 a-8}{4-a^{2}}
10. \dfrac{2 b^{2}+3 b-15}{b^{2}-49}-\dfrac{b^{2}+16 b-1}{49-b^{2}}
- Answer
-
\dfrac{3 b-2}{b+7}
11. \dfrac{8 y^{2}-10 y+7}{2 y-5}+\dfrac{2 y^{2}+7 y+2}{5-2 y}
In the following exercises, find the LCD.
12. \dfrac{7}{a^{2}-3 a-10}, \dfrac{3 a}{a^{2}-a-20}
- Answer
-
(a+2)(a-5)(a+4)
13. \dfrac{6}{n^{2}-4}, \dfrac{2 n}{n^{2}-4 n+4}
14. \dfrac{5}{3 p^{2}+17 p-6}, \dfrac{2 m}{3 p^{2}-23 p-8}
- Answer
-
(3 p+1)(p+6)(p+8)
In the following exercises, perform the indicated operations and simplify results.
15. \dfrac{7}{5 a}+\dfrac{3}{2 b}
16. \dfrac{2}{c-2}+\dfrac{9}{c+3}
- Answer
-
\dfrac{11 c-12}{(c-2)(c+3)}
17. \dfrac{3 x}{x^{2}-9}+\dfrac{5}{x^{2}+6 x+9}
18. \dfrac{2 x}{x^{2}+10 x+24}+\dfrac{3 x}{x^{2}+8 x+16}
- Answer
-
\dfrac{5 x^{2}+26 x}{(x+4)(x+4)(x+6)}
19. \dfrac{5 q}{p^{2} q-p^{2}}+\dfrac{4 q}{q^{2}-1}
20. \dfrac{3 y}{y+2}-\dfrac{y+2}{y+8}
- Answer
-
\dfrac{2\left(y^{2}+10 y-2\right)}{(y+2)(y+8)}
21. \dfrac{-3 w-15}{w^{2}+w-20}-\dfrac{w+2}{4-w}
22. \dfrac{7 m+3}{m+2}-5
- Answer
-
\dfrac{2 m-7}{m+2}
23. \dfrac{n}{n+3}+\dfrac{2}{n-3}-\dfrac{n-9}{n^{2}-9}
24. \dfrac{8 a}{a^{2}-64}-\dfrac{4}{a+8}
- Answer
-
\dfrac{4}{a-8}
25. \dfrac{5}{12 x^{2} y}+\dfrac{7}{20 x y^{3}}
Simplifying Complex Rational Expressions
In the following exercises, simplify.
1. \dfrac{\dfrac{7 x}{x+2}}{\dfrac{14 x^{2}}{x^{2}-4}}
- Answer
-
\dfrac{x-2}{2 x}
2. \dfrac{\dfrac{2}{5}+\dfrac{5}{6}}{\dfrac{1}{3}+\dfrac{1}{4}}
3. \dfrac{x-\dfrac{3 x}{x+5}}{\dfrac{1}{x+5}+\dfrac{1}{x-5}}
- Answer
-
\dfrac{(x-8)(x-5)}{2}
4. \dfrac{\dfrac{2}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}
5. \dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}
- Answer
-
\dfrac{11}{8}
6. \dfrac{\dfrac{3}{a^{2}}-\dfrac{1}{b}}{\dfrac{1}{a}+\dfrac{1}{b^{2}}}
7. \dfrac{\dfrac{2}{z^{2}-49}+\dfrac{1}{z+7}}{\dfrac{9}{z+7}+\dfrac{12}{z-7}}
- Answer
-
\dfrac{z-5}{21 z+21}
8. \dfrac{\dfrac{3}{y^{2}-4 y-32}}{\dfrac{2}{y-8}+\dfrac{1}{y+4}}
Solving Rational Equations
In the following exercises, solve.
1. \dfrac{1}{2}+\dfrac{2}{3}=\dfrac{1}{x}
- Answer
-
x=\dfrac{6}{7}
2. 1-\dfrac{2}{m}=\dfrac{8}{m^{2}}
3. \dfrac{1}{b-2}+\dfrac{1}{b+2}=\dfrac{3}{b^{2}-4}
- Answer
-
b=\dfrac{3}{2}
4. \dfrac{3}{q+8}-\dfrac{2}{q-2}=1
5. \dfrac{v-15}{v^{2}-9 v+18}=\dfrac{4}{v-3}+\dfrac{2}{v-6}
- Answer
-
no solution
6. \dfrac{z}{12}+\dfrac{z+3}{3 z}=\dfrac{1}{z}
In the following exercises, solve for the indicated variable.
7. \dfrac{V}{l}=h w for l
- Answer
-
l=\dfrac{V}{h w}
8. \dfrac{1}{x}-\dfrac{2}{y}=5 for y
9. x=\dfrac{y+5}{z-7} for z
- Answer
-
z=\dfrac{y+5+7 x}{x}
10. P=\dfrac{k}{V} for V