\( \begin{array}{llll}6 \text { a } \frac{1}{2 x}+\frac{3}{x} & \text { b } \frac{2}{3 y}+\frac{1}{y} & \text { c } \frac{3}{4 x}+\frac{1}{2 x} & \text { d } \frac{3}{x y}+\frac{2}{y} \\ \text { 41) } 7 \text { a } \frac{x}{4}+\frac{x+1}{2} & \text { b } \frac{x+2}{3}+\frac{x+7}{4} & \text { c } \frac{x-8}{4}+\frac{x+2}{5} & \text { d } \frac{x-4}{3}-\frac{x-5}{7}\end{array} \)

Sure, let's simplify each of the given expressions step by step. --- ### **Problem 6** **a.** \[ \frac{1}{2x} + \frac{3}{x} \] **Simplification:** \[ \frac{1}{2x} + \frac{3}{x} = \frac{1}{2x} + \frac{6}{2x} = \frac{7}{2x} \] --- **b.** \[ \frac{2}{3y} + \frac{1}{y} \] **Simplification:** \[ \frac{2}{3y} + \frac{1}{y} = \frac{2}{3y} + \frac{3}{3y} = \frac{5}{3y} \] --- **c.** \[ \frac{3}{4x} + \frac{1}{2x} \] **Simplification:** \[ \frac{3}{4x} + \frac{1}{2x} = \frac{3}{4x} + \frac{2}{4x} = \frac{5}{4x} \] --- **d.** \[ \frac{3}{xy} + \frac{2}{y} \] **Simplification:** \[ \frac{3}{xy} + \frac{2}{y} = \frac{3}{xy} + \frac{2x}{xy} = \frac{3 + 2x}{xy} \] --- ### **Problem 41** **a.** \[ \frac{x}{4} + \frac{x+1}{2} \] **Simplification:** \[ \frac{x}{4} + \frac{x+1}{2} = \frac{x}{4} + \frac{2(x+1)}{4} = \frac{x + 2x + 2}{4} = \frac{3x + 2}{4} \] --- **b.** \[ \frac{x+2}{3} + \frac{x+7}{4} \] **Simplification:** \[ \frac{x+2}{3} + \frac{x+7}{4} = \frac{4(x+2)}{12} + \frac{3(x+7)}{12} = \frac{4x + 8 + 3x + 21}{12} = \frac{7x + 29}{12} \] --- **c.** \[ \frac{x-8}{4} + \frac{x+2}{5} \] **Simplification:** \[ \frac{x-8}{4} + \frac{x+2}{5} = \frac{5(x-8)}{20} + \frac{4(x+2)}{20} = \frac{5x - 40 + 4x + 8}{20} = \frac{9x - 32}{20} \] --- **d.** \[ \frac{x-4}{3} - \frac{x-5}{7} \] **Simplification:** \[ \frac{x-4}{3} - \frac{x-5}{7} = \frac{7(x-4)}{21} - \frac{3(x-5)}{21} = \frac{7x - 28 - 3x + 15}{21} = \frac{4x - 13}{21} \] --- If you have any further questions or need additional explanations for any of the steps, feel free to ask!