# 4.10E: Exercises

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## Solve Rational Inequalities

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

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

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

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

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

$$[-1,3)$$

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

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

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

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

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

$$(-5,6)$$

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

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

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

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

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

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

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

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

$$[-9,6)$$

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

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

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

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

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

$$a=10$$

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

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

$$(1,4)$$

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

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

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

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

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

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

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

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

$$a=10$$

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

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

$$(1,4)$$

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

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

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

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

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

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

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

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

$$(-4,4)$$

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

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

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

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.

$$(2,5]$$

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.

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

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.

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