1. \(9, \; 1\)
2. \(-\frac{4}{3}, \; 2\)
3. \( \frac{7}{2}\)
4. \(\frac{11}{4}\)
5. \(-1\)
6. \(\frac{5}{3}, \, 3\)
7. \(-6, \, 2\)
8. \(\frac{1}{4}, \, 1\)
9. \(-2, \, \frac{8}{3}\)
10. \((1, \, 7)\)
11. \(\left( -\frac{16}{3}, \, \frac{14}{3} \right)\)
12. \(\left(-\frac{3}{2}, \, \frac{9}{2} \right)\)
13. \( \left[ -2, \, \frac{2}{3} \right]\)
14. \(\left( -\infty, \, -\frac{3}{5} \right] \cup \left[ -\frac{1}{5}, \, +\infty \right)\)
15. \((-\infty, \, -1) \cup \left( -\frac{1}{2}, \, +\infty \right)\)
16. \(\left( -\infty, \, -\frac{5}{2} \right] \cup \left[ \frac{25}{2} +\infty \right)\)
17. \([-\infty, \, -3] \cup [-1, \, 1] \cup [3, \, +\infty)\)
18. \([-4, \, -1) \cup (1, \, 4]\)
19. \((2, \, 4) – \{ 3 \}\)
20. \(\left( \frac{1}{2}, \, +\infty \right)\)
21. \(\left[\frac{2}{3}, \, 4 \right]\)
22. \([-1, \, 2] – \left\lbrace \frac{1}{2} \right\rbrace\)
23. \((-\infty, \, 1) \cup (2, \, +\infty)\)
24. \([-2, \, 2]\)
25. \((-1, \, 0) \cup (0, \, +\infty)\)
26. \(( -\infty, \, -7 ] \cup \left[ \frac{1}{3}, \, +\infty \right)\)
27. \(M = 43\)
28. \(M = 9\)
29. \(M = 10\)

En los problemas del 1 al 9, resolver la ecuación dada.

1. \[ \mid x – 5 \mid = 4 \]
2. \[ \mid 2x + 1 \mid = x + 3 \]
3. \[ \mid x – 2 \mid = 3x – 9 \]
4. \[ \mid x-2 \mid = 9-3x \]
5. \[ \mid x + 4 \mid = \mid 2 – x \mid \]
6. \[ \mid x – 1 \mid = \mid 2x -4 \mid \]
7. \[ \left| \frac{3x-2}{2} \right| = \mid x-4 \mid \]
8. \[ \left| 5-\frac{2}{x} \right| = 3 \]
9. \[ \left| \frac{x-5}{2x-3} \right| = 1 \]

En los problemas del 10 al 26, resolver la inecuación dada.

10. \[ \mid x – 4 \mid < 3 \]
11. \[ \mid 3x + 1 \mid < 15 \]
12. \[ \left| \frac{2x}{3} -1 \right| < 2 \]
13. \[ \mid -3x – 2 \mid \leq 4 \]
14. \[ \mid 5x + 2 \mid \geq 1 \]
15. \[ \mid -4x – 3 \mid > 1 \]
16. \[ \left| \frac{2x}{5}-2 \right| \geq 3 \]
17. \[ \mid x^2 – 5 \mid \geq 4 \]
18. \[ 1 < \mid x \mid \leq 4 \]
19. \[ 0 < \mid x – 3 \mid < 1 \]
20. \[ \mid x – 1 \mid < \mid x \mid \]
21. \[ \left| \frac{3-2x}{1+x} \right| \leq 1 \]
22. \[ \left| \frac{1}{1-2x} \right| \geq \frac{1}{3} \]
23. \[ \mid x – 1 \mid + \mid x – 2 \mid > 1 \]
24. \[ \mid x – 1 \mid + \mid x + 1 \mid \leq 4 \]
25. \[ \left| \frac{1}{2+x} \right| < \frac{1}{\mid x \mid} \]
26. \[ \mid 3x – 5 \mid \leq \mid 2x – 1 \mid + \mid 2x + 3 \mid \]

En los problemas del 27 al 29, hallar un número \(\boldsymbol{M}\) que satisfaga la proposición dada.

27. \[ \mid x + 2 \mid < 1 \Rightarrow \mid x^3 -x^2 + 2x + 1 \mid < M \]
    \[ \begin{aligned} &\mid x + 2 \mid < 1 \\[1em] &\hspace{2em} \Rightarrow \mid x^3 -x^2 + 2x + 1 \mid < M \end{aligned} \]
28. \[ \mid x – 3 \mid < 1/2 \Rightarrow \frac{\mid x+2 \mid}{\mid x-2 \mid} < M \]
    \[ \begin{aligned} &\mid x – 3 \mid < 1/2 \\[1em] &\hspace{5em}\Rightarrow \frac{\mid x+2 \mid}{\mid x-2 \mid} < M \end{aligned} \]
29. \[ \mid x – 1/4 \mid < 1/8 \Rightarrow \frac{\mid 16x+4 \mid}{1+x^2} < M \]
    \[ \begin{aligned} &\mid x – 1/4 \mid < 1/8 \\[1em] &\hspace{5em}\Rightarrow \frac{\mid 16x+4 \mid}{1+x^2} < M \end{aligned} \]
30. Probar las siguientes expresiones:
    1. \[ \mid x – y \mid \geq \mid x \mid – \mid y \mid \]

       Sugerencia: Aplicar la desigualdad triangular en:   \(x = (x – y) + y\).

    \[ \mid x – y \mid \geq \mid y \mid – \mid x \mid \]
    \[ \left| \mid x \mid – \mid y \mid \right| \leq \mid x – y \mid \]