7.7E: Ejercicios
- Page ID
- 112757
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Resolver desigualdades racionales
En los siguientes ejercicios, resolver cada desigualdad racional y escribir la solución en notación de intervalos.
1. \(\dfrac{x-3}{x+4} \geq 0\)
- Contestar
-
\((-\infty,-4) \cup[3, \infty)\)
2. \(\dfrac{x+6}{x-5} \geq 0\)
3. \(\dfrac{x+1}{x-3} \leq 0\)
- Contestar
-
\([-1,3)\)
4. \(\dfrac{x-4}{x+2} \leq 0\)
5. \(\dfrac{x-7}{x-1}>0\)
- Contestar
-
\((-\infty, 1) \cup(7, \infty)\)
6. \(\dfrac{x+8}{x+3}>0\)
7. \(\dfrac{x-6}{x+5}<0\)
- Contestar
-
\((-5,6)\)
8. \(\dfrac{x+5}{x-2}<0\)
9. \(\dfrac{3 x}{x-5}<1\)
- Contestar
-
\(\left(-\dfrac{5}{2}, 5\right)\)
10. \(\dfrac{5 x}{x-2}<1\)
11. \(\dfrac{6 x}{x-6}>2\)
- Contestar
-
\((-\infty,-3) \cup(6, \infty)\)
12. \(\dfrac{3 x}{x-4}>2\)
13. \(\dfrac{2 x+3}{x-6} \leq 1\)
- Contestar
-
\([-9,6)\)
14. \(\dfrac{4 x-1}{x-4} \leq 1\)
15. \(\dfrac{3 x-2}{x-4} \geq 2\)
- Contestar
-
\((-\infty,-6] \cup(4, \infty)\)
16. \(\dfrac{4 x-3}{x-3} \geq 2\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Contestar
-
\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Contestar
-
\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
21. \(\dfrac{2}{2 x^{2}+x-15} \geq 0\)
- Contestar
-
\((-\infty,-3) \cup\left(\dfrac{5}{2}, \infty\right)\)
22. \(\dfrac{6}{3 x^{2}-2 x-5} \geq 0\)
23. \(\dfrac{-2}{6 x^{2}-13 x+6} \leq 0\)
- Contestar
-
\(\left(-\infty, \dfrac{2}{3}\right) \cup\left(\dfrac{3}{2}, \infty\right)\)
24. \(\dfrac{-1}{10 x^{2}+11 x-6} \leq 0\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Contestar
-
\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Contestar
-
\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
25. \(\dfrac{1}{2}+\dfrac{12}{x^{2}}>\dfrac{5}{x}\)
- Contestar
-
\((-\infty, 0) \cup(0,4) \cup(6, \infty)\)
26. \(\dfrac{1}{3}+\dfrac{1}{x^{2}}>\dfrac{4}{3 x}\)
27. \(\dfrac{1}{2}-\dfrac{4}{x^{2}} \leq \dfrac{1}{x}\)
- Contestar
-
\([-2,0) \cup(0,4]\)
28. \(\dfrac{1}{2}-\dfrac{3}{2 x^{2}} \geq \dfrac{1}{x}\)
29. \(\dfrac{1}{x^{2}-16}<0\)
- Contestar
-
\((-4,4)\)
30. \(\dfrac{4}{x^{2}-25}>0\)
31. \(\dfrac{4}{x-2} \geq \dfrac{3}{x+1}\)
- Contestar
-
\([-10,-1) \cup(2, \infty)\)
32. \(\dfrac{5}{x-1} \leq \dfrac{4}{x+2}\)
Resolver una desigualdad con funciones racionales
En los siguientes ejercicios, resolver cada desigualdad de función racional y escribir la solución en notación de intervalos.
33. Dada la función\(R(x)=\dfrac{x-5}{x-2}\), encuentra los valores de\(x\) que hacen que la función sea menor o igual a 0.
- Contestar
-
\((2,5]\)
34. Dada la función\(R(x)=\dfrac{x+1}{x+3}\), encuentra los valores de\(x\) que hacen que la función sea menor o igual a 0.
35. Dada la función\(R(x)=\dfrac{x-6}{x+2}\), encuentra los valores de\(x\) que hacen que la función sea menor o igual a 0.
- Contestar
-
\((-\infty,-2) \cup[6, \infty)\)
36. Dada la función\(R(x)=\dfrac{x+1}{x-4}\), encuentra los valores de\(x\) que hacen que la función sea menor o igual a 0.
Ejercicios de escritura
37. Escribe los pasos que usarías para explicarle a tu hermano pequeño la solución de desigualdades racionales.
- Contestar
-
Las respuestas variarán
38. Crear una desigualdad racional cuya solución sea\((-\infty,-2] \cup[4, \infty)\).