In the exercises, from 1 to 21, solve the inequation. Construct the graph of 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 changed according to the interval \(5 ≤ C ≤ 20\). In what interval did the Fahrenheit temperature change on that day?
23. One day, the Fahrenheit temperature of a city changed according to the interval \(59 ≤ F ≤ 95\). In what interval did the Celsius temperature change 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 is 52 \(cm\). and 42 \(cm\). respectively.

    The machine cuts, from each corner of the sheets, equal-sized squares whose sides are \(x\, cm\). Then the machine shapes the metal into an open box by folding the sides up. Find the highest length \(x\) of the side of the squares if the area of the base of the box is at least 1200 \(cm^2\).

In the exercises, from 25 to 30, prove the proposition.

25. \[ a < b \wedge c > d \Rightarrow a - c < b - d \]
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 is between these numbers. 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 is between these numbers. 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: \(0 \leq (a - b)^2\).