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

$a \neq \dfrac{2}{3}$

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 $V$