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3.8e: Exercises - Polynomial and Rational Inequalities

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    A: Concepts

    Exercise \(\PageIndex{A}\) 

    1. Does the sign chart for any given polynomial or rational function always alternate? Explain and illustrate your answer with some examples.

    2. Write down your own steps for solving a rational inequality and illustrate them with an example. Do your steps also work for a polynomial inequality? Explain.

    Answer 1:

    1. Answer may vary

    B: Solve Polynomial Inequalities

    Exercise \(\PageIndex{B}\) 

    \( \bigstar\) Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation.

    3. \(x(x+1)(x-3) \geq 0\)

    4. \(x(x-1)(x+4) \geq 0\)

    5. \((x+2)(x-5)^{2}<0\)

    6. \((x-4)(x+1)^{2} \geq 0\)

    7. \((2 x-1)(x+3)(x+2) \leq 0\)

    8. \((3 x+2)(x-4)(x-5) \geq 0\)

    9. \(x(x+2)(x-5)^{2}<0\)

    10. \(x(2 x-5)(x-1)^{2}>0\)

    11. \(x(4 x+3)(x-1)^{2} \geq 0\)

    12. \((x-1)(x+1)(x-4)^{2}<0\)

    13. \((x+5)(x-10)(x-5)^{2} \geq 0\)

    14. \((3 x-1)(x-2)(x+2)^{2} \leq 0\)

    15. \(-4 x(4 x+9)(x-8)^{2}>0\)

    16. \(-x(x-10)(x+7)^{2}>0\)

    17. \(x^{3}+2 x^{2}-24 x \geq 0\)

    18. \(x^{3}-3 x^{2}-18 x \leq 0\)

    19. \(4 x^{3}-22 x^{2}-12 x<0\)

    20. \(9 x^{3}+30 x^{2}-24 x>0\)

    21. \(12 x^{4}+44 x^{3}>80 x^{2}\)

    22. \(6 x^{4}+12 x^{3}<48 x^{2}\)

    23. \(x\left(x^{2}+25\right)<10 x^{2}\)

    24. \(x^{3}>12 x(x-3)\)

    25. \(x^{4}-5 x^{2}+4 \leq 0\)

    26. \(x^{4}-13 x^{2}+36 \geq 0\)

    27. \(x^{4}>3 x^{2}+4\)

    28. \(4 x^{4}<3-11 x^{2}\)

    29. \(9 x^{3}-3 x^{2}-81 x+27 \leq 0\)

    30. \(2 x^{3}+x^{2}-50 x-25 \geq 0\)

    31. \(x^{3}-3 x^{2}+9 x-27>0\)

    32. \(3 x^{3}+5 x^{2}+12 x+20<0\)
    Answers 3-31:

    3. \( [-1,0] \cup [3, \infty)\)

    5. \((-\infty,-2)\)

    7. \((-\infty,-3] \cup\left[-2, \frac{1}{2}\right]\)

    9. \((-2,0)\)

    11. \(\left(-\infty,-\frac{3}{4}\right] \cup[0, \infty)\)

    13. \((-\infty,-5] \cup[5,5] \cup[10, \infty)\)

    15. \(\left(-\frac{9}{4}, 0\right)\)

    17. \([-6,0] \cup[4, \infty)\)

    19. \(\left(-\infty,-\frac{1}{2}\right) \cup(0,6)\)

    21. \((-\infty,-5) \cup\left(\frac{4}{3}, \infty\right)\)

    23. \((-\infty, 0)\)

    25. \([-2,-1] \cup[1,2]\)

    27. \((-\infty,-2) \cup(2, \infty)\)

    29. \((-\infty,-3] \cup\left[\frac{1}{3}, 3\right]\)

    31. \((3, \infty)\)
     

    C: Solve Rational Inequalities

    Exercise \(\PageIndex{C}\) 

    \( \bigstar\) Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation.

    33. \(\dfrac{x}{x-3} \ge 0 \\[6pt]\)

    34. \(\dfrac{x-5}{x} \ge 0\\[6pt]\)

    35. \(\dfrac{(x-3)(x+1)}{x}<0\\[6pt]\)

    36. \(\dfrac{(x+5)(x+4)}{(x-2)}<0\\[6pt]\)

    37. \(\dfrac{(2 x+1)(x+5)}{(x-3)(x-5)} \leq 0\\[6pt]\)

    38. \(\dfrac{(3 x-1)(x+6)}{(x-1)(x+9)} \geq 0\\[6pt]\)

    39. \(\dfrac{(x-8)(x+8)}{-2 x(x-2)} \geq 0\\[6pt]\)

    40. \(\dfrac{(2 x+7)(x+4)}{x(x+5)} \leq 0\\[6pt]\)

    41. \(\dfrac{x^{2}}{(2 x+3)(2 x-3)} \leq 0\\[6pt]\)

    42. \(\dfrac{(x-4)^{2}}{-x(x+1)}>0\\[6pt]\)

    43. \(\dfrac{-5 x(x-2)^{2}}{(x+5)(x-6)} \geq 0\\[6pt]\)

    44. \(\dfrac{(3 x-4)(x+5)}{x(x-4)^{2}} \geq 0\\[6pt]\)

    45. \(\dfrac{1}{(x-5)^{4}}>0\\[6pt]\)

    46. \(\dfrac{1}{(x-5)^{4}}<0\\[6pt]\)

    47. \(\dfrac{x^{2}-11 x-12}{x+4}<0\\[6pt]\)

    48. \(\dfrac{x^{2}-10 x+24}{x-2}>0\\[6pt]\)

    49. \(\dfrac{x^{2}+x-30}{2 x+1} \geq 0\\[6pt]\)

    50. \(\dfrac{2 x^{2}+x-3}{x-3} \leq 0\\[6pt]\)

    51. \(\dfrac{3 x^{2}-4 x+1}{x^{2}-9} \leq 0\\[6pt]\)

    52. \(\dfrac{x^{2}-16}{2 x^{2}-3 x-2} \geq 0\\[6pt]\)

    53. \(\dfrac{x^{2}-12 x+20}{x^{2}-10 x+25}>0\\[6pt]\)

    54. \(\dfrac{x^{2}+15 x+36}{x^{2}-8 x+16}<0\\[6pt]\)

    55. \(\dfrac{8 x^{2}-2 x-1}{2 x^{2}-3 x-14} \leq 0\\[6pt]\)

    56. \(\dfrac{4 x^{2}-4 x-15}{x^{2}+4 x-5} \geq 0\\[6pt]\)

    57. \(\dfrac{1}{x+5}+\dfrac{5}{x-1}>0\\[6pt]\)

    58. \(\dfrac{5}{x+4}-\dfrac{1}{x-4}<0\\[6pt]\)

    59. \(\dfrac{1}{x+7}>1\\[6pt]\)

    60. \(\dfrac{1}{x-1}<-5\\[6pt]\)

    61. \(x \geq \dfrac{30}{x-1}\\[6pt]\)

    62. \(x \leq \dfrac{1-2 x}{x-2}\\[6pt]\)

    63. \(\dfrac{1}{x-1} \leq \dfrac{2}{x}\\[6pt]\)

    64. \(\dfrac{3}{x+1}>-\dfrac{1}{x}\\[6pt]\)

    65. \(\dfrac{4}{x-3} \leq \dfrac{1}{x+3}\\[6pt]\)

    66. \(\dfrac{2 x-9}{x}+\dfrac{49}{x-8}<0\\[6pt]\)

    67. \(\dfrac{x}{2(x+2)}-\dfrac{1}{x+2} \leq \dfrac{12}{x(x+2)}\\[6pt]\)

    68. \(\dfrac{1}{2 x+1}-\dfrac{9}{2 x-1} \ge 2\\[6pt]\)

    69. \(\dfrac{3 x}{x^{2}-4}-\dfrac{2}{x-2}<0\\[6pt]\)

    70. \(\dfrac{x}{2 x+1}+\dfrac{4}{\\[6pt]2 x^{2}-7 x-4}<0\)

    71. \(\dfrac{x+1}{2 x^{2}+5 x-3} \geq \dfrac{x}{4 x^{2}-1}\\[6pt]\)

    72. \(\dfrac{x^{2}-14}{2 x^{2}-7 x-4} \leq \dfrac{5}{1+2 x}\\[6pt]\)
    Answers 33-71:

    33. \((-\infty,-0] \cup(3, \infty ) \)

    35. \((-\infty,-1) \cup(0,3)\)

    37. \(\left[-5,-\frac{1}{2}\right] \cup(3,5)\)

    39. \([-8,0) \cup(2,8]\)

    41. \(\left(-\frac{3}{2}, \frac{3}{2}\right)\)

    43. \((-\infty,-5) \cup[0,6)\)

    45. \((-\infty, 5) \cup(5, \infty)\)

    47. \((-\infty,-4) \cup(-1,12)\)

    49. \(\left[-6,-\frac{1}{2}\right) \cup[5, \infty)\)

    51. \(\left(-3, \frac{1}{3}\right] \cup[1,3)\)

    53. \((-\infty, 2) \cup(10, \infty)\)

    55. \(\left(-2,-\frac{1}{4}\right] \cup\left[\frac{1}{2}, \frac{7}{2}\right)\)

    57. \((-5,-4) \cup(1, \infty)\)

    59. \((-7,-6)\)

    61. \([-5,1) \cup[6, \infty)\)

    63. \((0,1) \cup[2, \infty)\)

    65. \((-\infty, 5] \cup(-3,3)\)

    67. \([-4,-2) \cup(0,6]\)

    69. \((-\infty,-2) \cup(2,4)\)

    71. \(\left(-3,-\frac{1}{2}\right) \cup\left(\frac{1}{2}, \infty\right)\)

    3.8e: Exercises - Polynomial and Rational Inequalities is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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