4.8.2E: Exercises

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

1. $\dfrac{x-3}{x+4} \geq 0$

Answer

$(-\infty,-4) \cup[3, \infty)$

2. $\dfrac{x+6}{x-5} \geq 0$

3. $\dfrac{x+1}{x-3} \leq 0$

Answer

$[-1,3)$

4. $\dfrac{x-4}{x+2} \leq 0$

5. $\dfrac{x-7}{x-1}>0$

Answer

$(-\infty, 1) \cup(7, \infty)$

6. $\dfrac{x+8}{x+3}>0$

7. $\dfrac{x-6}{x+5}<0$

Answer

$(-5,6)$

8. $\dfrac{x+5}{x-2}<0$

9. $\dfrac{3 x}{x-5}<1$

Answer

$\left(-\dfrac{5}{2}, 5\right)$

10. $\dfrac{5 x}{x-2}<1$

11. $\dfrac{6 x}{x-6}>2$

Answer

$(-\infty,-3) \cup(6, \infty)$

12. $\dfrac{3 x}{x-4}>2$

13. $\dfrac{2 x+3}{x-6} \leq 1$

Answer

$[-9,6)$

14. $\dfrac{4 x-1}{x-4} \leq 1$

15. $\dfrac{3 x-2}{x-4} \geq 2$

Answer

$(-\infty,-6] \cup(4, \infty)$

16. $\dfrac{4 x-3}{x-3} \geq 2$

17. $\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}$

Answer

$a=10$

18. $\dfrac{1}{x^{2}-4 x-12}>0$

19. $\dfrac{3}{x^{2}-5 x+4}<0$

Answer

$(1,4)$

20. $\dfrac{4}{x^{2}+7 x+12}<0$

21. $\dfrac{2}{2 x^{2}+x-15} \geq 0$

Answer

$(-\infty,-3) \cup\left(\dfrac{5}{2}, \infty\right)$

22. $\dfrac{6}{3 x^{2}-2 x-5} \geq 0$

23. $\dfrac{-2}{6 x^{2}-13 x+6} \leq 0$

Answer

$\left(-\infty, \dfrac{2}{3}\right) \cup\left(\dfrac{3}{2}, \infty\right)$

24. $\dfrac{-1}{10 x^{2}+11 x-6} \leq 0$

17. $\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}$

Answer

$a=10$

18. $\dfrac{1}{x^{2}-4 x-12}>0$

19. $\dfrac{3}{x^{2}-5 x+4}<0$

Answer

$(1,4)$

20. $\dfrac{4}{x^{2}+7 x+12}<0$

25. $\dfrac{1}{2}+\dfrac{12}{x^{2}}>\dfrac{5}{x}$

Answer

$(-\infty, 0) \cup(0,4) \cup(6, \infty)$

26. $\dfrac{1}{3}+\dfrac{1}{x^{2}}>\dfrac{4}{3 x}$

27. $\dfrac{1}{2}-\dfrac{4}{x^{2}} \leq \dfrac{1}{x}$

Answer

$[-2,0) \cup(0,4]$

28. $\dfrac{1}{2}-\dfrac{3}{2 x^{2}} \geq \dfrac{1}{x}$

29. $\dfrac{1}{x^{2}-16}<0$

Answer

$(-4,4)$

30. $\dfrac{4}{x^{2}-25}>0$

31. $\dfrac{4}{x-2} \geq \dfrac{3}{x+1}$

Answer

$[-10,-1) \cup(2, \infty)$

32. $\dfrac{5}{x-1} \leq \dfrac{4}{x+2}$

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation.

33. Given the function $R(x)=\dfrac{x-5}{x-2}$, find the values of $x$ that make the function less than or equal to 0.

Answer

$(2,5]$

34. Given the function $R(x)=\dfrac{x+1}{x+3}$, find the values of $x$ that make the function less than or equal to 0.

35. Given the function $R(x)=\dfrac{x-6}{x+2}$, find the values of $x$ that make the function less than or equal to 0.

Answer

$(-\infty,-2) \cup[6, \infty)$

36. Given the function $R(x)=\dfrac{x+1}{x-4}$, find the values of $x$ that make the function less than or equal to 0.

Writing Exercises

37. Write the steps you would use to explain solving rational inequalities to your little brother.

Answer

Answers will vary

38. Create a rational inequality whose solution is $(-\infty,-2] \cup[4, \infty)$.