\( \begin{array}{lll}\text { (1) } \frac{2 x}{3}+\frac{1}{6} & \text { (6) } \frac{5}{2 x}-\frac{x+1}{x-2} & \text { (10) } \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} \\ \begin{array}{lll}\text { (2) } \frac{7}{6 x}-\frac{2}{9 x y}+\frac{1}{3 x^{3}} & \text { (7) } \frac{2}{x}+\frac{5}{x+1} & \text { (1i) } \frac{2}{2 x y}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y+1}{4} \\ \text { (3) } \frac{3}{x}+\frac{2}{x^{2}} & \text { (8) } \frac{x+2}{x^{2}+2}-\frac{6}{x+2} & \text { (12) } \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \\ \text { (4) } 1+a-\frac{2 a-1}{2} & \text { (9) } \frac{x-3}{x+3}-\frac{x+3}{x-3} & \text { (3) } x-\frac{2 x}{3 x-2} \\ \text { (5) } \frac{5}{6 x y}-\frac{x+2}{2 x} & & \end{array}\end{array} \)

To solve the given expressions, we will simplify each one step by step. Let's go through them one by one. ### Expression (1) \[ \frac{2x}{3} + \frac{1}{6} \] To combine these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. \[ \frac{2x}{3} = \frac{2x \cdot 2}{3 \cdot 2} = \frac{4x}{6} \] Now we can add: \[ \frac{4x}{6} + \frac{1}{6} = \frac{4x + 1}{6} \] ### Expression (2) \[ \frac{7}{6x} - \frac{2}{9xy} + \frac{1}{3x^3} \] The common denominator for \(6x\), \(9xy\), and \(3x^3\) is \(18x^3y\). Rewriting each term: \[ \frac{7}{6x} = \frac{7 \cdot 3x^2y}{18x^3y} = \frac{21x^2y}{18x^3y} \] \[ \frac{2}{9xy} = \frac{2 \cdot 2x^2}{18x^3y} = \frac{4x^2}{18x^3y} \] \[ \frac{1}{3x^3} = \frac{1 \cdot 6y}{18x^3y} = \frac{6y}{18x^3y} \] Now combine: \[ \frac{21x^2y - 4x^2 + 6y}{18x^3y} = \frac{(21y - 4)x^2 + 6y}{18x^3y} \] ### Expression (3) \[ \frac{3}{x} + \frac{2}{x^2} \] The common denominator is \(x^2\): \[ \frac{3}{x} = \frac{3x}{x^2} \] Now combine: \[ \frac{3x + 2}{x^2} \] ### Expression (4) \[ 1 + a - \frac{2a - 1}{2} \] Rewriting: \[ 1 + a - \frac{2a - 1}{2} = 1 + a - (a - \frac{1}{2}) = 1 + \frac{1}{2} = \frac{3}{2} \] ### Expression (5) \[ \frac{5}{6xy} - \frac{x + 2}{2x} \] The common denominator is \(6xy\): \[ \frac{5}{6xy} - \frac{(x + 2) \cdot 3y}{6xy} = \frac{5 - 3y(x + 2)}{6xy} \] ### Expression (6) \[ \frac{5}{2x} - \frac{x + 1}{x - 2} \] The common denominator is \(2x(x - 2)\): \[ \frac{5(x - 2) - 2x(x + 1)}{2x(x - 2)} = \frac{5x - 10 - 2x^2 - 2x}{2x(x - 2)} = \frac{-2x^2 + 3x - 10}{2x(x - 2)} \] ### Expression (7) \[ \frac{2}{x} + \frac{5}{x + 1} \] The common denominator is \(x(x + 1)\): \[ \frac{2(x + 1) + 5x}{x(x + 1)} = \frac{2x + 2 + 5x}{x(x + 1)} = \frac{7x + 2}{x(x + 1)} \] ### Expression (8) \[ \frac{x + 2}{x^2 + 2} - \frac{6}{x + 2} \] The common denominator is \((x^2 + 2)(x + 2)\): \[ \frac{(x + 2)^2 - 6(x^2 + 2)}{(x^2 + 2)(x + 2)} = \frac{x^2 + 4x + 4 - 6x^2 - 12}{(x^2 + 2)(x + 2)} = \frac{-5x^2 + 4x - 8}{(x^2 + 2)(x + 2)} \] ### Expression (9) \[ \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} \] The common denominator is \((x + 3)(x - 3)\): \[ \frac{(x - 3)^2 - (x + 3)^2}{(x + 3)(x - 3)} = \frac{x^2 - 6x + 9 - (x^2 + 6x + 9)}{(x + 3)(x - 3)} = \frac{-12x}{(x + 3)(x - 3)} \] ### Expression (10) \[ \frac{4}{(2x + 1)^2} - \frac{x + 1}{2x + 1} \] The common denominator is \((2x + 1)^2\): \[ \frac{4 - (x + 1)(2x + 1)}{(2x + 1)^2} = \frac{4 - (2x^2 + 3x + 1)}{(2x + 1)^2} = \frac{-2x^2 - 3x + 3}{(2x + 1)^2} \] ### Expression (11) \[ \frac{x - 3}{3} - \frac{x - 2}{2} + \frac{x + 1}{6} \] The common denominator is 6: \[ \frac{2(x - 3) - 3(x - 2) + (x + 1)}{6} = \frac{2x - 6 - 3x + 6 + x + 1}{6} = \frac{0x + 1}{6} = \frac{1}{6} \] Now we have simplified all the expressions. If you need further assistance or specific calculations, please let me know!