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
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\((-\infty,-4) \cup[3, \infty)\)
2. \(\dfrac{x+6}{x-5} \geq 0\)
3. \(\dfrac{x+1}{x-3} \leq 0\)
- Answer
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\([-1,3)\)
4. \(\dfrac{x-4}{x+2} \leq 0\)
5. \(\dfrac{x-7}{x-1}>0\)
- Answer
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\((-\infty, 1) \cup(7, \infty)\)
6. \(\dfrac{x+8}{x+3}>0\)
7. \(\dfrac{x-6}{x+5}<0\)
- Answer
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\((-5,6)\)
8. \(\dfrac{x+5}{x-2}<0\)
9. \(\dfrac{3 x}{x-5}<1\)
- Answer
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\(\left(-\dfrac{5}{2}, 5\right)\)
10. \(\dfrac{5 x}{x-2}<1\)
11. \(\dfrac{6 x}{x-6}>2\)
- Answer
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\((-\infty,-3) \cup(6, \infty)\)
12. \(\dfrac{3 x}{x-4}>2\)
13. \(\dfrac{2 x+3}{x-6} \leq 1\)
- Answer
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\([-9,6)\)
14. \(\dfrac{4 x-1}{x-4} \leq 1\)
15. \(\dfrac{3 x-2}{x-4} \geq 2\)
- Answer
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\((-\infty,-6] \cup(4, \infty)\)
16. \(\dfrac{4 x-3}{x-3} \geq 2\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Answer
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\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Answer
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\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
21. \(\dfrac{2}{2 x^{2}+x-15} \geq 0\)
- Answer
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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. \(\dfrac{-2}{6 x^{2}-13 x+6} \leq 0\)
- Answer
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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. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Answer
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\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Answer
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\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
25. \(\dfrac{1}{2}+\dfrac{12}{x^{2}}>\dfrac{5}{x}\)
- Answer
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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. \(\dfrac{1}{2}-\dfrac{4}{x^{2}} \leq \dfrac{1}{x}\)
- Answer
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\([-2,0) \cup(0,4]\)
28. \(\dfrac{1}{2}-\dfrac{3}{2 x^{2}} \geq \dfrac{1}{x}\)
29. \(\dfrac{1}{x^{2}-16}<0\)
- Answer
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\((-4,4)\)
30. \(\dfrac{4}{x^{2}-25}>0\)
31. \(\dfrac{4}{x-2} \geq \dfrac{3}{x+1}\)
- Answer
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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.
- Answer
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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.
- Answer
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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.
