4.3: Unit 3 Practice Answers

MTH 95: Algebra 3 · LibreTexts

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    Unit 3 Answers

    Section 3.1P: Direct and Inverse Variation

    1. \(400\)
    2. \(1682\)
    3. \(\dfrac{209}{8}\) or \(26.125\)
    4. \(\dfrac{56}{25}\) or \(2.24\)
    5. \(\dfrac{16}{3}\)
    6. \(\dfrac{250}{27}\)
    7. a. Let d be the distance the spring stretches, and w be the weight on the spring. Then \(d = 8w\)
      b. \(40\) inches
    8. a. Let I be the intensity of light and d the distance from the light source. Then \(I = \dfrac{320}{d^2}\)
      b. \(0.99\) foot-candles
    9. a. Let n be the number of people who sign up and P be the price per person. Then \(P = \dfrac{3672}{n}\)
      b. \(\$104.91\)

    Section 3.2P: Graphs of Rational Functions

    1. Vertical asymptote: \(x = -1\)
    2. Vertical asymptote: \(x = -2\)
    3. Vertical asymptote: \(x = 4\)
    4. Vertical asymptote: \(x = 0\)
    5. Horizontal asymptote: \(y = 9\)
    6. Horizontal asymptote: \(y = -1\)
    7. Horizontal asymptote: \(y = 0\)
    8. D = {x: \(x \ne 0\)}, R = {y: \(y \ne 0\)}
    9. D = {x: \(x \ne 4\)}, R = {y: \(y \ne 0\)}
    10. D = {x: \(x \ne 0\)}, R = {y: \(y \ne -2\)}
    11. D = {x: \(x \ne 2\)}, R = {y: \(y \ne -3\)}
    12. D = {x: \(x \ne 3\)}, R = {y: \(y \ne -4\)}

    Section 3.3P: Reducing Rational Expressions

    1. \(\dfrac{x-9}{5}\); \(x \ne 1\)
    2. \(\dfrac{x+5}{x}\); \(x \ne 7\)
    3. \(-1\); \(x \ne -2\)
    4. \(-2\); \(x \ne 3\)
    5. \(-3x\); \(x \ne -2\)
    6. \(-\dfrac{x+4}{x+5}\); \(x \ne 5\)
    7. \(-\dfrac{x-5}{9}\); \(x \ne -2\)
    8. \(\dfrac{x(x+6)}{(x+9)(x+4)}\); \(x \ne −7\)
    9. \(−\dfrac{(x+2)(x−9)}{(x+1)(x+4)}\); \(x \ne 1, −5\)
    10. \(\dfrac{(x−3)(x−5)}{(3x−2)(x−8)}\); \(x \ne 2, −5\)
    11. \(−\dfrac{(x+6)(x−3)}{(2x+3)(x+3)}\); \(x \ne 5\)
    12. \(−\dfrac{(x−1)(x−6)}{(x−3)(x+5)}\); \(x \ne -1, 4, 6\)
    13. \(\dfrac{(x−5)(x−3)}{(5x−2)(x+1)}\); \(x \ne -5, 3\)
    14. \(−\dfrac{(x−5)(x+8)}{(x−4)(x+9)}\); \(x \ne −8, −6\)
    15. \(\dfrac{(x−7)(x−3)}{(7x−2)(x+3)}\); \(x \ne −2, 3, 4\)

    Section 3.4P: Sums and Differences of Rational Functions

    1. 7(x−1); \(x \ne 6\)
    2. 9(x−7); \(x \ne 1\)
    3. 4(x−8); \(x \ne 7\)
    4. 3(x−8); \(x \ne 2\)
    5. \(\dfrac{3(x+1)}{(x−1)(x−9)}\); \(x \ne 5\)
    6. \(\dfrac{5(x+14)}{x(x−14)}\); \(x \ne 7\)
    7. \(\dfrac{−7(x+3)}{(x−3)(x−7)}\); \(x \ne 5\)
    8. \(\dfrac{−9(x+4)}{x(x−4)}\); \(x \ne 2\)
    9. \(\dfrac{−8}{x+10}\);  \(x \ne 5, 0, −10\)

    Section 3.5P: Solving Rational Equations

    1. \(x = −\dfrac{7}{2}\)
      See graph here for checking your solution: https://www.desmos.com/calculator/p2zs8clqmf
    2. \(x = 0\)
    3. No solution
    4. \(x = 41\)
    5. \(x = −\dfrac{128}{9}\)
    6. \(x = \dfrac{7+\sqrt{97}}{8}, \dfrac{7−\sqrt{97}}{8}\)
    7. \(x = −7, −12 \)
    8. \(\dfrac{−19+\sqrt{73}}{4}, \dfrac{−19−\sqrt{73}}{4}\)
    9. \(x = 2\) (note that \(x = 3\) is an extraneous solution)
    10. \(x = \dfrac{5−\sqrt{17}}{2}, \dfrac{−5−\sqrt{17}}{2}\)
    11. \(x = \dfrac{7}{2}, −\dfrac{4}{3}\)
    12. \(x = 3\) (note that \(x = 2\) is an extraneous solution)
    13. \(x = 4, \dfrac{3}{2}\)
    14. \(x = \dfrac{7}{2}, \dfrac{5}{2}\)
    15. \((-1, 3)\)
    16. \((-1, \infty)\)
    17. \((-\infty, -6] \cup (4, \infty)\)

    Section 3.6P: Rational Applications

    1. 15 hours
    2. \(\dfrac{28}{11}\) hours
    3. 4.90 mph
    4. 15 mph
    5. 19 barrels

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