Exercises

Precalculus for Everybody

Chapter 1. Preliminaries

Absolute Value

In the exercises, from 1 to 9, solve the equation.

  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. \[ \left| \frac{x-5}{2x-3} \right| = 1 \]

In the exercises, from 10 to 26, solve the inequation.

  1. \[ \mid x - 4 \mid < 3 \]
  2. \[ \mid 3x + 1 \mid < 15 \]
  3. \[ \left| \frac{2x}{3} -1 \right| < 2 \]
  4. \[ \mid -3x - 2 \mid \leq 4 \]
  5. \[ \mid 5x + 2 \mid \geq 1 \]
  6. \[ \mid -4x - 3 \mid > 1 \]
  7. \[ \left| \frac{2x}{5}-2 \right| \geq 3 \]
  8. \[ \mid x^2 - 5 \mid \geq 4 \]
  9. \[ 1 < \mid x \mid \leq 4 \]
  10. \[ 0 < \mid x - 3 \mid < 1 \]
  11. \[ \mid x - 1 \mid < \mid x \mid \]
  12. \[ \left| \frac{3-2x}{1+x} \right| \leq 1 \]
  13. \[ \left| \frac{1}{1-2x} \right| \geq \frac{1}{3} \]
  14. \[ \mid x - 1 \mid + \mid x - 2 \mid > 1 \]
  15. \[ \mid x - 1 \mid + \mid x + 1 \mid \leq 4 \]
  16. \[ \left| \frac{1}{2+x} \right| < \frac{1}{\mid x \mid} \]
  17. \[ \mid 3x - 5 \mid \leq \mid 2x - 1 \mid + \mid 2x + 3 \mid \]

In the exercises, from 27 to 29, find a number \(\boldsymbol{M}\) that satisfies the given inequality.

  1. \[ \mid x + 2 \mid < 1 \Rightarrow \mid x^3 -x^2 + 2x + 1 \mid < M \]
    \[ \begin{aligned} &\mid x + 2 \mid < 1 \\[1em] &\hspace{2em} \Rightarrow \mid x^3 -x^2 + 2x + 1 \mid < M \end{aligned} \]
  2. \[ \mid x - 3 \mid < 1/2 \Rightarrow \frac{\mid x+2 \mid}{\mid x-2 \mid} < M \]
    \[ \begin{aligned} &\mid x - 3 \mid < 1/2 \\[1em] &\hspace{5em}\Rightarrow \frac{\mid x+2 \mid}{\mid x-2 \mid} < M \end{aligned} \]
  3. \[ \mid x - 1/4 \mid < 1/8 \Rightarrow \frac{\mid 16x+4 \mid}{1+x^2} < M \]
    \[ \begin{aligned} &\mid x - 1/4 \mid < 1/8 \\[1em] &\hspace{5em}\Rightarrow \frac{\mid 16x+4 \mid}{1+x^2} < M \end{aligned} \]
  4. Prove:
    1. \[ \mid x - y \mid \geq \mid x \mid - \mid y \mid \]

      Suggestion: Apply the triangle inequality to \(x = (x - y) + y\).

    2. \[ \mid x - y \mid \geq \mid y \mid - \mid x \mid \]
    3. \[ \left| \mid x \mid - \mid y \mid \right| \leq \mid x - y \mid \]
    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. \[ M = 10 \]