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# 7.8: 7.E Review Exercises and Sample Exam

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## Review Exercises

Exercise $$\PageIndex{1}$$ Simplifying Rational Expressions

Evaluate for the given set of $$x$$-values.

1. $$\frac{25}{2x^{2}}$$; {$$−5, 0, 5$$}
2. $$\frac{x−4}{2x−1}$$; {$$\frac{1}{2}, 2, 4$$}
3. $$\frac{1}{x^{2}+9}$$; {$$−3, 0, 3$$}
4. $$\frac{x+3}{x^{2}−9}$$; {$$−3, 0, 3$$}

1. $$\frac{1}{2}$$, undefined, $$\frac{1}{2}$$

3. $$\frac{1}{18}, \frac{1}{9}, \frac{1}{18}$$

Exercise $$\PageIndex{2}$$ Simplifying Rational Expressions

State the restrictions to the domain.

1. $$\frac{5}{x}$$
2. $$\frac{1}{x(3x+1)}$$
3. $$\frac{x+2}{x^{2}−25}$$
4. $$\frac{x−1}{(x−1)(2x−3)}$$

1. $$x≠0$$

3. $$x≠±5$$

Exercise $$\PageIndex{3}$$ Simplifying Rational Expressions

State the restrictions and simplify.

1. $$\frac{x−8}{x^{2}−64}$$
2. $$\frac{3x^{2}+9x}{2x^{3}−18x}$$
3. $$\frac{x^{2}−5x−24}{x^{2}−3x−40}$$
4. $$\frac{2x^{2}+9x−5}{4x^{2}−1}$$
5. $$\frac{x^{2}−144}{12−x}$$
6. $$\frac{8x^{2}−10x−3}{9−4x^{2}}$$
7. Given $$f(x)=\frac{x−3}{x^{2}+9}$$, find $$f(−3), f(0)$$, and $$f(3)$$.
8. Simplify $$g(x)=\frac{x^{2}−2x−24}{2x^{2}−9x−18}$$ and state the restrictions.

1. $$\frac{1}{x+8}$$; $$x≠±8$$

3. $$\frac{x+3}{x+5}$$; $$x≠−5, 8$$

5. $$−(x+12)$$; $$x≠12$$

7. $$f(−3)=−\frac{1}{3}, f(0)=−\frac{1}{3}, f(3)=0$$

Exercise $$\PageIndex{4}$$ Multiplying and Dividing Rational Expressions

Multiply. (Assume all denominators are nonzero.)

1. $$\frac{3x^{5}}{x−3}\cdot\frac{x−3}{9x^{2}}$$
2. $$\frac{12y^{2}}{y^{3}(2y−1)}\cdot\frac{(2y−1)}{3y}$$
3. $$\frac{3x^{2}}{x−2}\cdot\frac{x^{2}−4x+4}{5x^{3}}$$
4. $$\frac{x^{2}−8x+15}{9x^{5}}\cdot\frac{12x^{2}}{x−3}$$
5. $$\frac{x^{2}−36}{x^{2}−x−30}\cdot\frac{2x^{2}+10x}{x^{2}+5x−6}$$
6. $$\frac{9x^{2}+11x+2}{4−81x^{2}}\cdot\frac{9x−2}{(x+1)^{2}}$$

1. $$\frac{x^{3}}{3}$$

3. $$\frac{3(x−2)}{5x}$$

5. $$\frac{2x}{x−1}$$

Exercise $$\PageIndex{5}$$ Multiplying and Dividing Rational Expressions

Divide. (Assume all denominators are nonzero.)

1. $$\frac{9x^{2}−25}{5x^{3}}\div\frac{3x+5}{15x^{4}}$$
2. $$\frac{4x^{2}}{4x^{2}−1}\div\frac{2x^{2}}{x−1}$$
3. $$\frac{3x^{2}−13x−10}{x^{2}−x−20}\div\frac{9x^{2}+12x+4}{x^{2}+8x+16}$$
4. $$\frac{2x^{2}+xy−y^{2}}{x^{2}+2xy+y^{2}}\div\frac{4x^{2}−y^{2}}{3x^{2}+2xy−y^{2}}$$
5. $$\frac{2x^{2}−6x−20}{8x^{2}+17x+2}\div (8x^{2}−39x−5)$$
6. $$\frac{12x^{2}−27x^{4}}{15x^{4}+10x^{3}}\div (3x^{2}+x−2)$$
7. $$\frac{25y^{2}−15y}{4(y−2)}\cdot\frac{1}{5y−1}\div \frac{10y}{2(y−2)^{2}}$$
8. $$\frac{10x^{4}}{1−36x^{2}}\div\frac{5x^{2}}{6x^{2}−7x+1}\cdot x−12x$$
9. Given $$f(x)=\frac{1}{6x^{2}−9x+5}$$ and $$g(x)=\frac{x^{2}+3x−10}{4x^{2}+5x−6}$$, calculate $$(f⋅g)(x)$$ and state the restrictions.
10. Given $$f(x)=\frac{x+7}{5x−1}$$ and $$g(x)=\frac{x^{2}−49}{25x^{2}−5x}$$, calculate $$(f/g)(x)$$ and state the restrictions.

1. $$3x(3x−5)$$

3. $$\frac{x+4}{3x+2}$$

5. $$\frac{2}{(8x+1)^{2}}$$

7. $$\frac{5y^{2}-13y+6}{4(5y-1)}$$

9. $$(f⋅g)(x)=\frac{(4x+3)(x−2)}{x+2}$$; $$x≠−5, −2, 34$$

Exercise $$\PageIndex{6}$$ Adding and Subtracting Rational Expressions

Simplify. (Assume all denominators are nonzero.)

1. $$\frac{5x}{y}−\frac{3}{y}$$
2. $$\frac{x}{x^{2}−x−6}−\frac{3}{x^{2}−x−6}$$
3. $$\frac{2x}{2x+1}+\frac{1}{x−5}$$
4. $$\frac{3}{x−7}+\frac{1−2x}{x^{2}}$$
5. $$\frac{7x}{4x^{2}−9x+2}−\frac{2}{x−2}$$
6. $$\frac{5}{x−5}+\frac{20−9x}{2x^{2}−15x+25}$$
7. $$\frac{x}{x−5}−\frac{2}{x−3}−\frac{5(x−3)}{x^{2}−8x+15}$$
8. $$\frac{3x}{2x−1}−\frac{x−4}{x+4}+\frac{12(2−x)}{2x^{2}+7x−4}$$
9. $$\frac{1}{x^{2}+8x−9}−\frac{1}{x^{2}+11x+18}$$
10. $$\frac{4}{x^{2}+13x+36}+\frac{3}{x^{2}+6x−27}$$
11. $$\frac{y+1}{y+2}−\frac{1}{2−y}+\frac{2y}{y^{2}−4}$$
12. $$\frac{1}{y−11}−\frac{y−2}{y^{2}−1}$$
13. Given $$f(x)=x+12x−5$$ and $$g(x)=\frac{x}{x+1}$$, calculate $$(f+g)(x)$$ and state the restrictions.
14. Given $$f(x)=x+13x$$ and $$g(x)=\frac{2}{x−8}$$, calculate $$(f−g)(x)$$ and state the restrictions.

1. $$\frac{5x−3}{y}$$

3. $$\frac{2x^{2}−8x+1}{(2x+1)(x−5)}$$

5. $$−\frac{1}{4x−1}$$

7. $$\frac{x−5}{x−3}$$

9. $$\frac{3}{(x−1)(x+2)(x+9)}$$

11. $$\frac{y}{y−2}$$

13. $$(f+g)(x)=\frac{3x^{2}−3x+1}{(2x−5)(x+1)}$$; $$x≠−1, \frac{5}{2}$$

Exercise $$\PageIndex{7}$$ Complex Fractions

Simplify.

1. $$\frac{4−\frac{2}{x}}{ \frac{2x−1}{3x}}$$
2. $$\frac{\frac{1}{3}−\frac{1}{3y}}{\frac{1}{5}−\frac{1}{5y}}$$
3. $$\frac{\frac{1}{6}+\frac{1}{x}}{\frac{1}{36}-\frac{1}{x^{2}}}$$
4. $$\frac{\frac{1}{100}−\frac{1}{x^{2}}}{\frac{1}{10}−\frac{1}{x}}$$
5.  $$\frac{\frac{x}{x+3}−\frac{2}{x+1}}{ \frac{ x}{x+4}+\frac{1}{x+3}}$$
6.  $$\frac{\frac{3x−1}{x−5}}{ 5x+2−\frac{2}{x}}$$
7. $$\frac{1−12x+35x^{2} }{1−25x^{2}}$$
8. $$2−15x+\frac{25x^{2}}{2x−5}$$

1. $$6$$

3. $$\frac{6x}{x−6}$$

5. $$\frac{(x−3)(x+4)}{(x+1)(x+2)}$$

7. $$-\frac{7x-1}{5x+1}$$

Exercise $$\PageIndex{8}$$ Solving Rational Equations

Solve.

1. $$\frac{6}{x−6}=\frac{2}{2x−1}$$
2. $$\frac{x}{x−6}=\frac{x+2}{x−2}$$
3. $$\frac{1}{3x}-\frac{2}{9}=\frac{1}{x}$$
4. $$\frac{2}{x−5}+\frac{3}{5}=\frac{1}{x−5}$$
5. $$\frac{x}{x−5}+\frac{4}{x+5}=−10x^{2}−25$$
6. $$\frac{2x−12}{2x+3}=\frac{2−3x^{2}}{2x^{2}+3x}$$
7. $$\frac{x+1}{2(x−2)}+\frac{x−6}{x}=1$$
8. $$\frac{5x+2}{x+1}−\frac{x}{x+4}=4$$
9. $$\frac{x}{x+5}+\frac{1}{x−4}=\frac{4x−7}{x^{2}+x−20}$$
10. $$\frac{2}{3x−1}+\frac{x}{2x+1}=\frac{2(3−4x)}{6x^{2}+x−1}$$
11. $$\frac{x}{x−1}+\frac{1}{x+1}=\frac{2x}{x^{2}−1}$$
12. $$\frac{2x}{x+5}−\frac{1}{2x−3}=\frac{4−7x^{2}}{x^{2}+7x−15}$$
13. Solve for $$a$$: $$\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$$.
14. Solve for $$y$$: $$x=\frac{2}{y}−\frac{1}{3y}$$.

1. $$−\frac{3}{5}$$

3. $$−3$$

5. $$−10, 1$$

7. $$3, 8$$

9. $$3$$

11. $$Ø$$

13. $$a=\frac{bc}{b+c}$$

Exercise $$\PageIndex{9}$$ Applications of Rational Equations

Use algebra to solve the following applications.

1. A positive integer is twice another. The sum of the reciprocals of the two positive integers is $$\frac{1}{4}$$. Find the two integers.
2. If the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is $$\frac{3}{10}$$. Find the integers.
3. Mary can jog, on average, $$2$$ miles per hour faster than her husband, James. James can jog $$6.6$$ miles in the same amount of time it takes Mary to jog $$9$$ miles. How fast, on average, can Mary jog?
4. Billy traveled $$140$$ miles to visit his grandmother on the bus and then drove the $$140$$ miles back in a rental car. The bus averages $$14$$ miles per hour slower than the car. If the total time spent traveling was $$4.5$$ hours, then what was the average speed of the bus?
5. Jerry takes twice as long as Manny to assemble a skateboard. If they work together, they can assemble a skateboard in $$6$$ minutes. How long would it take Manny to assemble the skateboard without Jerry’s help?
6. Working alone, Joe completes the yard work in $$30$$ minutes. It takes Mike $$45$$ minutes to complete work on the same yard. How long would it take them working together?

1. $$6, 12$$

3. $$7.5$$ miles per hour

5. $$9$$ minutes

Exercise $$\PageIndex{10}$$ Variation

Construct a mathematical model given the following.

1. $$y$$ varies directly with $$x$$, and $$y = 12$$ when $$x = 4$$.
2. $$y$$ varies inversely as $$x$$, and $$y = 2$$ when $$x = 5$$.
3. $$y$$ is jointly proportional to $$x$$ and $$z$$, where $$y = 36$$ when $$x = 3$$ and $$z = 4$$.
4. $$y$$ is directly proportional to the square of $$x$$ and inversely proportional to $$z$$, where $$y = 20$$ when $$x = 2$$ and $$z = 5$$.
5. The distance an object in free fall drops varies directly with the square of the time that it has been falling. It is observed that an object falls $$16$$ feet in $$1$$ second. Find an equation that models the distance an object will fall and use it to determine how far it will fall in $$2$$ seconds.
6. The weight of an object varies inversely as the square of its distance from the center of earth. If an object weighs $$180$$ pounds on the surface of earth (approximately $$4,000$$ miles from the center), then how much will it weigh at $$2,000$$ miles above earth’s surface?

1. $$y=3x$$

3. $$y=3xz$$

5. $$d=16t^{2}$$; $$64$$ feet

## Sample Exam

Exercise $$\PageIndex{11}$$

Simplify and state the restrictions.

1. $$\frac{15x^{3}(3x−1)^{2}}{3x(3x−1)}$$
2. $$\frac{x^{2}−144}{x^{2}+12x}$$
3. $$\frac{x^{2}+x−12}{2x^{2}+7x−4}$$
4. $$\frac{9−x^{2}}{(x−3)^{2}}$$

1. $$\frac{5x^{2}(3x−1)^{2}}{x-1}$$; $$x≠1$$

3. $$\frac{x−3}{2x−1}$$; $$x≠−4, \frac{1}{2}$$

Exercise $$\PageIndex{12}$$

Simplify. (Assume all variables in the denominator are positive.)

1. $$\frac{5x}{x^{2}−25}\cdot\frac{x−5}{25x^{2}}$$
2. $$\frac{x^{2}+x−6}{x^{2}−4x+4}\cdot\frac{3x^{2}−5x−2}{x^{2}−9}$$
3. $$\frac{x^{2}−4x−12}{12x^{2}}\div\frac{x−6}{6x}$$
4. $$\frac{2x^{2}−7x−4}{6x^{2}−24x}\div\frac{2x^{2}+7x+3}{10x^{2}+30x}$$
5. $$\frac{1}{x−5}+\frac{1}{x+5}$$
6. $$\frac{x}{x+1}−\frac{8}{2−x}−\frac{12x}{x^{2}−x−2}$$
7. $$\frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y^{2}}-\frac{1}{x^{2}}}$$
8. $$\frac{1−6x+9x^{2}}{2−5x−3x^{2}}$$
9. Given $$f(x)=\frac{x^{2}−81}{(4x−3)^{2}}$$ and $$g(x)=\frac{4x−3}{x−9}$$, calculate $$(f⋅g)(x)$$ and state the restrictions.
10. Given $$f(x)=\frac{x}{x−5}$$ and $$g(x)=\frac{1}{3x−5}$$, calculate $$(f−g)(x)$$ and state the restrictions.

1. $$\frac{1}{5x(x+5)}$$

3. $$\frac{x+2}{2x}$$

5. $$\frac{2x}{(x−5)(x+5)}$$

7. $$\frac{xy}{x−y}$$

9. $$(f⋅g)(x)=\frac{x+9}{4x−3}$$; $$x≠\frac{3}{4}, 9$$

Exercise $$\PageIndex{13}$$

Solve.

1. $$\frac{1}{3}+\frac{1}{x}=2$$
2. $$\frac{1}{x−5}=\frac{3}{2x−3}$$
3. $$1−9x+20x^{2}=0$$
4. $$\frac{x+2}{x−2}+\frac{1}{x+2}=\frac{4(x+1)}{x^{2}−4}$$
5. $$\frac{x}{x−2}−\frac{1}{x−3}=\frac{3x−10}{x^{2}−5x+6}$$
6. $$\frac{5}{x+4}−\frac{x}{4−x}=\frac{9x−4}{x^{2}−16}$$
7. Solve for $$r$$:$$P=\frac{120}{1+3r}$$.

1. $$\frac{3}{5}$$

3. $$\frac{1}{4}, \frac{1}{5}$$

5. $$4$$

7. $$r=40P−\frac{1}{3}$$

Exercise $$\PageIndex{14}$$

Set up an algebraic equation and then solve.

1. An integer is three times another. The sum of the reciprocals of the two integers is $$\frac{1}{3}$$. Find the two integers.
2. Working alone, Joe can paint the room in $$6$$ hours. If Manny helps, then together they can paint the room in $$2$$ hours. How long would it take Manny to paint the room by himself?
3. A river tour boat averages $$6$$ miles per hour in still water. With the current, the boat can travel $$17$$ miles in the same time it can travel $$7$$ miles against the current. What is the speed of the current?
4. The breaking distance of an automobile is directly proportional to the square of its speed. Under optimal conditions, a certain automobile moving at $$35$$ miles per hour can break to a stop in $$25$$ feet. Find an equation that models the breaking distance under optimal conditions and use it to determine the breaking distance if the automobile is moving $$28$$ miles per hour.
2. $$3$$ hours
4. $$y=\frac{1}{49}x^{2}$$; $$16$$ feet