19.4E: Exercises
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Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
1. \(\dfrac{x-3}{x+4} \geq 0\)
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\((-\infty,-4) \cup[3, \infty)\)
2. \(\dfrac{x+6}{x-5} \geq 0\)
3. \(\dfrac{x+1}{x-3} \leq 0\)
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\([-1,3)\)
4. \(\dfrac{x-4}{x+2} \leq 0\)
5. \(\dfrac{x-7}{x-1}>0\)
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\((-\infty, 1) \cup(7, \infty)\)
6. \(\dfrac{x+8}{x+3}>0\)
7. \(\dfrac{x-6}{x+5}<0\)
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\((-5,6)\)
8. \(\dfrac{x+5}{x-2}<0\)
9. \(\dfrac{3 x}{x-5}<1\)
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\(\left(-\dfrac{5}{2}, 5\right)\)
10. \(\dfrac{5 x}{x-2}<1\)
11. \(\dfrac{6 x}{x-6}>2\)
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\((-\infty,-3) \cup(6, \infty)\)
12. \(\dfrac{3 x}{x-4}>2\)
13. \(\dfrac{2 x+3}{x-6} \leq 1\)
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\([-9,6)\)
14. \(\dfrac{4 x-1}{x-4} \leq 1\)
15. \(\dfrac{3 x-2}{x-4} \geq 2\)
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\((-\infty,-6] \cup(4, \infty)\)
16. \(\dfrac{4 x-3}{x-3} \geq 2\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
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\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
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\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
21. \(\dfrac{2}{2 x^{2}+x-15} \geq 0\)
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\((-\infty,-3) \cup\left(\dfrac{5}{2}, \infty\right)\)
22. \(\dfrac{6}{3 x^{2}-2 x-5} \geq 0\)
23. \(\dfrac{-2}{6 x^{2}-13 x+6} \leq 0\)
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\(\left(-\infty, \dfrac{2}{3}\right) \cup\left(\dfrac{3}{2}, \infty\right)\)
24. \(\dfrac{-1}{10 x^{2}+11 x-6} \leq 0\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
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\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
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\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
25. \(\dfrac{1}{2}+\dfrac{12}{x^{2}}>\dfrac{5}{x}\)
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\((-\infty, 0) \cup(0,4) \cup(6, \infty)\)
26. \(\dfrac{1}{3}+\dfrac{1}{x^{2}}>\dfrac{4}{3 x}\)
27. \(\dfrac{1}{2}-\dfrac{4}{x^{2}} \leq \dfrac{1}{x}\)
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\([-2,0) \cup(0,4]\)
28. \(\dfrac{1}{2}-\dfrac{3}{2 x^{2}} \geq \dfrac{1}{x}\)
29. \(\dfrac{1}{x^{2}-16}<0\)
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\((-4,4)\)
30. \(\dfrac{4}{x^{2}-25}>0\)
31. \(\dfrac{4}{x-2} \geq \dfrac{3}{x+1}\)
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\([-10,-1) \cup(2, \infty)\)
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.
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\((2,5]\)
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.
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\((-\infty,-2) \cup[6, \infty)\)
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.
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Answers will vary
38. Create a rational inequality whose solution is \((-\infty,-2] \cup[4, \infty)\).