1. Solve: \((2x + 1) + \cfrac{3}{(2x + 1)} = 4\), where \(x ≠ -\cfrac{1}{2}\)

2. Solve:
\(\cfrac{1}{(x-1)(x-2)}\) \(+\cfrac{1}{(x-2)(x-3)} \) \(+\cfrac{1}{(x-3)(x-4)}\) \(=\cfrac{1}{6} ,\) \(x≠1,2,3,4\).

3. Let us solve: \[ (2x + 1) + \frac{3}{2x + 1} = 4,\; x \ne -\frac{1}{2} \]

4. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

5. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

6. If \(x + \cfrac{1}{x} = 3\), then what is the value of \(\cfrac{(x^2 + 3x + 1)}{(x^2 + 7x + 1)}\)?

(a) \(\cfrac{3}{5}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{2}{4}\) (d) \(\cfrac{1}{2}\)

7. If \(a + \cfrac{1}{b} = 1\) and \(b + \cfrac{1}{c} = 1\), then find the values of \((c + \cfrac{1}{a})\) and \((abc + 1)\).

(a) 1 and 0 (b) 0 and 1 (c) 0 and 0 (d) 1 and 1

8. Let Anik's speed be \(x\) meters per second.
∴ Salma's speed = \((x+1)\) meters per second
∴ Time taken by Anik to cover 180 meters = \(\cfrac{180}{x}\) seconds
Time taken by Salma to cover 180 meters = \(\cfrac{180}{x+1}\) seconds

According to the question, \(\cfrac{180}{x} - \cfrac{180}{x+1} = 2\)
Or, \(\cfrac{180(x+1-x)}{x(x+1)} = 2\)
Or, \(\cfrac{180}{x^2 + x} = 2\)
Or, \(\cfrac{90}{x^2 + x} = 1\)
Or, \(x^2 + x = 90\)
Or, \(x^2 + x - 90 = 0\)
Or, \(x^2 + 10x - 9x - 90 = 0\)
Or, \(x(x + 10) - 9(x + 10) = 0\)
Or, \((x + 10)(x - 9) = 0\)

∴ Either \(x + 10 = 0\), so \(x = -10\),
Or, \(x - 9 = 0\), so \(x = 9\)
Since speed cannot be negative,

∴ Anik's speed is \(9\) meters per second.

9. If \(\cfrac{x}{y+z} = \cfrac{y}{z+x} = \cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or (-1).

10. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or (-1).

11. If \(\cfrac{a}{1-a}+\cfrac{b}{1-b}+\cfrac{c}{1-c}=1\), then prove that \(\cfrac{1}{1-a}+\cfrac{1}{1-b}+\cfrac{c}{1-c}=4\).

12. Let us solve: \((2x + 1)^2 + (x + 1)^2 = 6x + 47\)

13. Solve: \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6}, \quad x \ne 1,\ 2,\ 3,\ 4 \]