# 2.EC: Exercises for Rational Expressions and Equations

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## Simplifying, Multiplying, and Dividing Rational Expressions

In the following exercises, determine the values for which the rational expression is undefined.

1. $$\dfrac{5 a+3}{3 a-2}$$

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

2. $$\dfrac{b-7}{b^{2}-25}$$

3. $$\dfrac{5 x^{2} y^{2}}{8 y}$$

$$y \neq 0$$

4. $$\dfrac{x-3}{x^{2}-x-30}$$

In the following exercises, simplify.

5. $$\dfrac{9 m^{4}}{18 m n^{3}}$$

6. $$\dfrac{x^{2}+7 x+12}{x^{2}+8 x+16}$$

$$\dfrac{x+3}{x+4}$$

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. $$\dfrac{72 x-12 x^{2}}{8 x+32} \cdot \dfrac{x^{2}+10 x+24}{x^{2}-36}$$

$$\dfrac{-3 x}{2}$$

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. $$\dfrac{x^{2}-4 x-12}{x^{2}+8 x+12} \div \dfrac{x^{2}-36}{3 x}$$

$$\dfrac{3 x}{(x+6)(x+6)}$$

12. $$\dfrac{y^{2}-16}{4} \div \dfrac{y^{3}-64}{2 y^{2}+8 y+32}$$

13. $$\dfrac{11+w}{w-9} \div \dfrac{121-w^{2}}{9-w}$$

$$\dfrac{1}{11+w}$$

14. $$\dfrac{3 y^{2}-12 y-63}{4 y+3} \div\left(6 y^{2}-42 y\right)$$

15. $$\dfrac{\dfrac{c^{2}-64}{3 c^{2}+26 c+16}}{\dfrac{c^{2}-4 c-32}{15 c+10}}$$

$$\dfrac{5}{c+4}$$

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. $$\dfrac{y^{2}+10 y}{y+5}+\dfrac{25}{y+5}$$

$$y+5$$

3. $$\dfrac{7 x^{2}}{x^{2}-9}+\dfrac{21 x}{x^{2}-9}$$

4. $$\dfrac{x^{2}}{x-7}-\dfrac{3 x+28}{x-7}$$

$$x+4$$

5. $$\dfrac{y^{2}}{y+11}-\dfrac{121}{y+11}$$

6. $$\dfrac{4 q^{2}-q+3}{q^{2}+6 q+5}-\dfrac{3 q^{2}-q-6}{q^{2}+6 q+5}$$

$$\dfrac{q-3}{q+5}$$

7. $$\dfrac{5 t+4 t+3}{t^{2}-25}-\dfrac{4 t^{2}-8 t-32}{t^{2}-25}$$

8. $$\dfrac{18 w}{6 w-1}+\dfrac{3 w-2}{1-6 w}$$

$$\dfrac{15 w+2}{6 w-1}$$

9. $$\dfrac{a^{2}+3 a}{a^{2}-4}-\dfrac{3 a-8}{4-a^{2}}$$

10. $$\dfrac{2 b^{2}+3 b-15}{b^{2}-49}-\dfrac{b^{2}+16 b-1}{49-b^{2}}$$

$$\dfrac{3 b-2}{b+7}$$

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. $$\dfrac{7}{a^{2}-3 a-10}, \dfrac{3 a}{a^{2}-a-20}$$

$$(a+2)(a-5)(a+4)$$

13. $$\dfrac{6}{n^{2}-4}, \dfrac{2 n}{n^{2}-4 n+4}$$

14. $$\dfrac{5}{3 p^{2}+17 p-6}, \dfrac{2 m}{3 p^{2}-23 p-8}$$

$$(3 p+1)(p+6)(p+8)$$

In the following exercises, perform the indicated operations and simplify results.

15. $$\dfrac{7}{5 a}+\dfrac{3}{2 b}$$

16. $$\dfrac{2}{c-2}+\dfrac{9}{c+3}$$

$$\dfrac{11 c-12}{(c-2)(c+3)}$$

17. $$\dfrac{3 x}{x^{2}-9}+\dfrac{5}{x^{2}+6 x+9}$$

18. $$\dfrac{2 x}{x^{2}+10 x+24}+\dfrac{3 x}{x^{2}+8 x+16}$$

$$\dfrac{5 x^{2}+26 x}{(x+4)(x+4)(x+6)}$$

19. $$\dfrac{5 q}{p^{2} q-p^{2}}+\dfrac{4 q}{q^{2}-1}$$

20. $$\dfrac{3 y}{y+2}-\dfrac{y+2}{y+8}$$

$$\dfrac{2\left(y^{2}+10 y-2\right)}{(y+2)(y+8)}$$

21. $$\dfrac{-3 w-15}{w^{2}+w-20}-\dfrac{w+2}{4-w}$$

22. $$\dfrac{7 m+3}{m+2}-5$$

$$\dfrac{2 m-7}{m+2}$$

23. $$\dfrac{n}{n+3}+\dfrac{2}{n-3}-\dfrac{n-9}{n^{2}-9}$$

24. $$\dfrac{8 a}{a^{2}-64}-\dfrac{4}{a+8}$$

$$\dfrac{4}{a-8}$$

25. $$\dfrac{5}{12 x^{2} y}+\dfrac{7}{20 x y^{3}}$$

## Simplifying Complex Rational Expressions

In the following exercises, simplify.

1. $$\dfrac{\dfrac{7 x}{x+2}}{\dfrac{14 x^{2}}{x^{2}-4}}$$

$$\dfrac{x-2}{2 x}$$

2. $$\dfrac{\dfrac{2}{5}+\dfrac{5}{6}}{\dfrac{1}{3}+\dfrac{1}{4}}$$

3. $$\dfrac{x-\dfrac{3 x}{x+5}}{\dfrac{1}{x+5}+\dfrac{1}{x-5}}$$

$$\dfrac{(x-8)(x-5)}{2}$$

4. $$\dfrac{\dfrac{2}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}$$

5. $$\dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}$$

$$\dfrac{11}{8}$$

6. $$\dfrac{\dfrac{3}{a^{2}}-\dfrac{1}{b}}{\dfrac{1}{a}+\dfrac{1}{b^{2}}}$$

7. $$\dfrac{\dfrac{2}{z^{2}-49}+\dfrac{1}{z+7}}{\dfrac{9}{z+7}+\dfrac{12}{z-7}}$$

$$\dfrac{z-5}{21 z+21}$$

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. $$\dfrac{1}{2}+\dfrac{2}{3}=\dfrac{1}{x}$$

$$x=\dfrac{6}{7}$$

2. $$1-\dfrac{2}{m}=\dfrac{8}{m^{2}}$$

3. $$\dfrac{1}{b-2}+\dfrac{1}{b+2}=\dfrac{3}{b^{2}-4}$$

$$b=\dfrac{3}{2}$$

4. $$\dfrac{3}{q+8}-\dfrac{2}{q-2}=1$$

5. $$\dfrac{v-15}{v^{2}-9 v+18}=\dfrac{4}{v-3}+\dfrac{2}{v-6}$$

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 $$l$$

$$l=\dfrac{V}{h w}$$

8. $$\dfrac{1}{x}-\dfrac{2}{y}=5$$ for $$y$$

9. $$x=\dfrac{y+5}{z-7}$$ for $$z$$

$$z=\dfrac{y+5+7 x}{x}$$
10. $$P=\dfrac{k}{V}$$ for $$V$$