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Mathematics LibreTexts

7.8: 7.E Review Exercises and Sample Exam

  • Page ID
    22446
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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. \(\frac{x+3}{x^{2}−9}\); {\(−3, 0, 3\)}
    Answer

    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. \(\frac{x−1}{(x−1)(2x−3)}\)
    Answer

    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.
    Answer

    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. \(\frac{9x^{2}+11x+2}{4−81x^{2}}\cdot\frac{9x−2}{(x+1)^{2}}\)
    Answer

    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.
    Answer

    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.
    Answer

    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. \(2−15x+\frac{25x^{2}}{2x−5}\)
    Answer

    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 \(y\): \(x=\frac{2}{y}−\frac{1}{3y}\).
    Answer

    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?
    Answer

    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?
    Answer

    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. \(\frac{9−x^{2}}{(x−3)^{2}}\)
    Answer

    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.
    Answer

    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 \(r\):\( P=\frac{120}{1+3r}\).
    Answer

    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.
    Answer

    2. \(3\) hours

    4. \(y=\frac{1}{49}x^{2}\); \(16\) feet