1.6E: Exercises - Polynomial and Rational Inequalities
- Page ID
- 26490
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
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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.
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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:
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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.
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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:
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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. \(\left(-3,-\frac{1}{2}\right) \cup\left(\frac{1}{2}, \infty\right)\)