2.EC: Exercises for Rational Expressions and Equations
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
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\(a \neq \dfrac{2}{3}\)
2. \(\dfrac{b-7}{b^{2}-25}\)
3. \(\dfrac{5 x^{2} y^{2}}{8 y}\)
- Answer
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\(y \neq 0\)
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
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\(\dfrac{x+3}{x+4}\)
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
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\(\dfrac{-3 x}{2}\)
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
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\(\dfrac{3 x}{(x+6)(x+6)}\)
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
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\(\dfrac{1}{11+w}\)
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
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\(\dfrac{5}{c+4}\)
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
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\(y+5\)
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
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\(x+4\)
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
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\(\dfrac{q-3}{q+5}\)
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
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\(\dfrac{15 w+2}{6 w-1}\)
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
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\(\dfrac{3 b-2}{b+7}\)
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
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\((a+2)(a-5)(a+4)\)
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
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\((3 p+1)(p+6)(p+8)\)
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
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\(\dfrac{11 c-12}{(c-2)(c+3)}\)
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
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\(\dfrac{5 x^{2}+26 x}{(x+4)(x+4)(x+6)}\)
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
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\(\dfrac{2\left(y^{2}+10 y-2\right)}{(y+2)(y+8)}\)
21. \(\dfrac{-3 w-15}{w^{2}+w-20}-\dfrac{w+2}{4-w}\)
22. \(\dfrac{7 m+3}{m+2}-5\)
- Answer
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\(\dfrac{2 m-7}{m+2}\)
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
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\(\dfrac{4}{a-8}\)
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
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\(\dfrac{x-2}{2 x}\)
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
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\(\dfrac{(x-8)(x-5)}{2}\)
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
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\(\dfrac{11}{8}\)
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
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\(\dfrac{z-5}{21 z+21}\)
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
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\(x=\dfrac{6}{7}\)
2. \(1-\dfrac{2}{m}=\dfrac{8}{m^{2}}\)
3. \(\dfrac{1}{b-2}+\dfrac{1}{b+2}=\dfrac{3}{b^{2}-4}\)
- Answer
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\(b=\dfrac{3}{2}\)
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
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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
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\(l=\dfrac{V}{h w}\)
8. \(\dfrac{1}{x}-\dfrac{2}{y}=5\) for \(y\)
9. \(x=\dfrac{y+5}{z-7}\) for \(z\)
- Answer
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\(z=\dfrac{y+5+7 x}{x}\)
10. \(P=\dfrac{k}{V}\) for \(V\)