# 1.6E: 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.

### 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. $$3 x^{3}+5 x^{2}+12 x+20<0$$
 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. $$\dfrac{x^{2}-14}{2 x^{2}-7 x-4} \leq \dfrac{5}{1+2 x}\\[6pt]$$
 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. $$\left(-3,-\frac{1}{2}\right) \cup\left(\frac{1}{2}, \infty\right)$$