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Mathematics LibreTexts

7.7E: Exercises

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    30802
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    Solve Rational Inequalities

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

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

    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 \((-\infty,-2] \cup[4, \infty)\).


    7.7E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.