6. \( \left(\frac{x+1}{8}-\frac{2 x-3}{16}\right)=3\left(\frac{3}{4} x-\frac{1}{4}\right)-\frac{3}{8}(3 x-2) \)

To solve the equation \[ \left(\frac{x+1}{8}-\frac{2 x-3}{16}\right)=3\left(\frac{3}{4} x-\frac{1}{4}\right)-\frac{3}{8}(3 x-2), \] let's simplify both sides step by step. **Left Side:** 1. The left side is \(\frac{x+1}{8} - \frac{2x-3}{16}\). For the first term, we can express it as follows: \[ \frac{x+1}{8} = \frac{2(x+1)}{16} = \frac{2x + 2}{16}. \] So now we have: \[ \frac{2x + 2}{16} - \frac{2x - 3}{16} = \frac{(2x + 2) - (2x - 3)}{16} = \frac{2x + 2 - 2x + 3}{16} = \frac{5}{16}. \] So, the left side simplifies to \(\frac{5}{16}\). **Right Side:** 2. The right side is \(3\left(\frac{3}{4} x - \frac{1}{4}\right) - \frac{3}{8}(3x - 2)\). Starting with the first term: \[ 3\left(\frac{3}{4}x - \frac{1}{4}\right) = \frac{9}{4}x - \frac{3}{4}. \] Next, we simplify the second term: \[ \frac{3}{8}(3x - 2) = \frac{9x - 6}{8}. \] Now, we need a common denominator to combine the terms on the right side. The least common denominator between 4 and 8 is 8: \[ \frac{9}{4}x - \frac{3}{4} = \frac{18}{8}x - \frac{6}{8}. \] So the right side becomes: \[ \frac{18}{8}x - \frac{6}{8} - \frac{9x - 6}{8} = \frac{18}{8}x - \frac{6}{8} - \frac{9x}{8} + \frac{6}{8}. \] Now combining like terms: \[ \left(\frac{18}{8}x - \frac{9}{8}x\right) + \left(-\frac{6}{8} + \frac{6}{8}\right) = \frac{9}{8}x. \] **Set Left Side Equal to Right Side:** Now we equate both simplified sides: \[ \frac{5}{16} = \frac{9}{8}x. \] To solve for \(x\), we first multiply both sides by 16: \[ 5 = 18x. \] Finally, dividing both sides by 18 gives: \[ x = \frac{5}{18}. \] So, the solution to the equation is \[ \boxed{\frac{5}{18}}. \]