(12) \( \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \) (13) \( x-\frac{2 x}{3 x-2} \)

Simplify the expression by following steps: - step0: Solution: \(x-\frac{2x}{\left(3x-2\right)}\) - step1: Remove the parentheses: \(x-\frac{2x}{3x-2}\) - step2: Reduce fractions to a common denominator: \(\frac{x\left(3x-2\right)}{3x-2}-\frac{2x}{3x-2}\) - step3: Transform the expression: \(\frac{x\left(3x-2\right)-2x}{3x-2}\) - step4: Multiply the terms: \(\frac{3x^{2}-2x-2x}{3x-2}\) - step5: Subtract the terms: \(\frac{3x^{2}-4x}{3x-2}\) Calculate or simplify the expression \( (x-3)/3 - (x-2)/2 + (x+1)/6 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-3\right)}{3}-\frac{\left(x-2\right)}{2}+\frac{\left(x+1\right)}{6}\) - step1: Remove the parentheses: \(\frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x-3\right)\times 2}{3\times 2}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step3: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step4: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{6}+\frac{x+1}{6}\) - step5: Transform the expression: \(\frac{\left(x-3\right)\times 2-\left(x-2\right)\times 3+x+1}{6}\) - step6: Multiply the terms: \(\frac{2x-6-\left(x-2\right)\times 3+x+1}{6}\) - step7: Multiply the terms: \(\frac{2x-6-\left(3x-6\right)+x+1}{6}\) - step8: Calculate: \(\frac{1}{6}\) Let's simplify each expression step by step. ### Problem (12) The expression is: \[ \frac{x-3}{3} - \frac{x-2}{2} + \frac{x+1}{6} \] To simplify this, we need a common denominator. The least common multiple of the denominators \(3\), \(2\), and \(6\) is \(6\). 1. Rewrite each term with the common denominator: - \(\frac{x-3}{3} = \frac{2(x-3)}{6} = \frac{2x - 6}{6}\) - \(\frac{x-2}{2} = \frac{3(x-2)}{6} = \frac{3x - 6}{6}\) - \(\frac{x+1}{6} = \frac{x+1}{6}\) 2. Combine the fractions: \[ \frac{2x - 6 - (3x - 6) + (x + 1)}{6} \] 3. Simplify the numerator: \[ 2x - 6 - 3x + 6 + x + 1 = 0 + 1 = 1 \] Thus, the expression simplifies to: \[ \frac{1}{6} \] ### Problem (13) The expression is: \[ x - \frac{2x}{3x-2} \] To simplify this, we can combine the terms over a common denominator, which is \(3x - 2\). 1. Rewrite \(x\) with the common denominator: \[ x = \frac{x(3x-2)}{3x-2} = \frac{3x^2 - 2x}{3x - 2} \] 2. Combine the fractions: \[ \frac{3x^2 - 2x - 2x}{3x - 2} = \frac{3x^2 - 4x}{3x - 2} \] Thus, the expression simplifies to: \[ \frac{3x^2 - 4x}{3x - 2} \] ### Final Results - For (12): \(\frac{1}{6}\) - For (13): \(\frac{3x^2 - 4x}{3x - 2}\)