1. \((- \infty, \, 4)\)
2. \(\left( – \infty, \, -\frac{32}{3} \right)\)
3. \(\left( \frac{17}{2}, \, + \infty \right)\)
4. \( \left( – \infty , \, -\frac{43}{37} \right] \)
5. \( \left( \frac{17}{5}, \, 19 \right] \)
6. \((-19, \, -9)\)
7. \((-2, \, 3)\)
8. \((-1 , \, 1)\)
9. \(\left( – \infty, \, -1 – \sqrt{21} \right] \cup \left[ -1 + \sqrt{21}, \, +\infty \right) \)
10. \(( – \infty, \, -3 ) \cup \left( \frac{1}{2}, \, + \infty \right) \)
11. \(\left( – \infty, \, \frac{1}{3} \right) \cup \left( \frac{2}{3}, \, +\infty \right)\)
12. \((3, \, 4)\)
13. \([-3, \, -2] \cup [1, \, +\infty)\)
14. \(( -2, \, 2 ]\)
15. \(\left[ -\frac{10}{3}, \, 0 \right)\)
16. \(\left[ \frac{1}{3}, \, 1 \right) \)
17. \((-\infty, \, -2] \cup (0, \, 2]\)
18. \(\left( -\infty, \, -1 – \sqrt{3} \right] \cup \left( -1, \, -1 + \sqrt{3} \right]\)
19. \(( -3, \, -1 ] \cup ( 0, \, +\infty ) \)
20. \((-\infty, \, -2 ) \cup (1, \, +\infty)\)
21. \(\left( 2 – 2 \sqrt{3}, \, 0 \right) \cup \left( 3, \, 2 + 2 \sqrt{3} \right)\)
22. \(41 \leq F \leq 68\)
23. \(15 \leq C \leq 35\)
24. \(6 \; cm\)

For exercises 1 through 21, solve the inequality and graph the solution set.

1. \[ 4x – 5 < 2x + 3 \]
2. \[ 2(x – 5) – 3 > 5(x + 4) – 1 \]
3. \[ \frac{2x-5}{3} -3 > 1 \]
4. \[ \frac{5x-1}{4} – \frac{x+1}{3} \leq \frac{3x-13}{10} \]
5. \[ 8 \geq \frac{2x-5}{3} -3 > 1-x \]
6. \[ 5 < \frac{x-1}{-2} < 10 \]
7. \[ (x-3)(x+2) < 0 \]
8. \[ x^2-1 < 0 \]
9. \[ x^2+2x-20\geq 0 \]
10. \[ 2x^2+5x-3>0 \]
11. \[ 9x-2 < 9x^2 \]
12. \[ (x-2)(x-5) < -2 \]
13. \[ (x+2)(x-1)(x+3) \geq 0 \]
14. \[ \frac{x-2}{x+2} \leq 0 \]
15. \[ \frac{2}{x} \leq -\frac{3}{5} \]
16. \[ \frac{2}{x-1} \leq -3 \]
17. \[ \frac{x}{2} + \frac{1}{x} \leq \frac{3}{x} \]
18. \[ \frac{1}{x+1} – \frac{x-2}{3} \geq 1 \]
19. \[ \frac{x-1}{x+3} < \frac{x+2}{x} \]
20. \[ \frac{x+1}{1-x} < \frac{x}{2+x} \]
21. \[ \frac{4-2x}{x^2+2} > 2 – \frac{x}{x-3} \]
22. One day, the Celsius temperature of a city ranged from 5 to 20 degrees. What was the corresponding interval for the Fahrenheit temperature on that day?
23. One day, the Fahrenheit temperature of a city changed within the interval 59 ≤ F ≤ 95. What was the interval for the Celsius temperature on that day?

24. (Highest length) One machine produces open boxes using rectangular sheets of metal as raw material. The length and width of each sheet are 52 \(cm\) and 42 \(cm\) respectively.

    The machine cuts equal-sized squares with sides of length \(x\) \(cm\) from each corner of the sheets. It then shapes the metal into an open box by folding up the sides. Find the largest possible length x for the side of the squares, given that the area of the base of the box must be at least 1200 \(cm^2\).

In exercises 25 through 30, prove the proposition.

25. \[ a d \Rightarrow a – c < b – d \]
26. \[ a \neq 0 \Rightarrow a^2 > 0 \]
27. \[ a > 1 \Rightarrow a^2 > a \]
28. \[ 0 < a < 1 \Rightarrow a^2 < a \]
29. \[ 0 < a < b \wedge 0 < c < d \Rightarrow ac < bd \]
30. \(a \neq 0 \Rightarrow a\)   and   \(a^{-1}\) have the same sign (both positive or negative).
31. The arithmetic mean of two numbers, \(a\) and \(b\) is the number \(\frac{a+b}{2}\). Prove that the arithmetic mean of two numbers falls between them. This is, prove that:
    \[ a < b \Rightarrow a < \frac{a+b}{2} < b \]
32. The geometric mean of two positive numbers, \(a\) and \(b\), is the number \(\sqrt{ab}\). Prove that the geometric mean of two positive numbers falls between them. This is, prove that:
    \[ 0 < a < b \Rightarrow a <\sqrt{ab} < b \]
33. Prove that \(\sqrt{ab}\leq \frac{a+b}{2} \), where \(a \geq 0\) and \(b \geq 0\).   Hint: \(0 \leq (a – b)^2\).