Solve x-3/2-x-2/3/1-frac{1{6}}=frac{x/3-frac{x}{2}}{frac{1}{2}+frac{1}{3}}+1

$$\frac { \frac { x - 3 } { 2 } - \frac { x - 2 } { 3 } } { 1 - \frac { 1 } { 6 } } = \frac { \frac { x } { 3 } - \frac { x } { 2 } } { \frac { 1 } { 2 } + \frac { 1 } { 3 } } + 1$$

Solve for x

$$x=5$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{x-3}{2} times \frac{3}{3}. Multiply \frac{x-2}{3} times \frac{2}{2}.

$$\frac{\frac{3\left(x-3\right)}{6}-\frac{2\left(x-2\right)}{6}}{1-\frac{1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Since \frac{3\left(x-3\right)}{6} and \frac{2\left(x-2\right)}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3\left(x-3\right)-2\left(x-2\right)}{6}}{1-\frac{1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Do the multiplications in 3\left(x-3\right)-2\left(x-2\right).

$$\frac{\frac{3x-9-2x+4}{6}}{1-\frac{1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Combine like terms in 3x-9-2x+4.

$$\frac{\frac{x-5}{6}}{1-\frac{1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Convert 1 to fraction \frac{6}{6}.

$$\frac{\frac{x-5}{6}}{\frac{6}{6}-\frac{1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Since \frac{6}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-5}{6}}{\frac{6-1}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Subtract 1 from 6 to get 5.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=\frac{\frac{x}{3}-\frac{x}{2}}{\frac{1}{2}+\frac{1}{3}}+1$$

Combine \frac{x}{3} and -\frac{x}{2} to get -\frac{1}{6}x.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=\frac{-\frac{1}{6}x}{\frac{1}{2}+\frac{1}{3}}+1$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=\frac{-\frac{1}{6}x}{\frac{3}{6}+\frac{2}{6}}+1$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=\frac{-\frac{1}{6}x}{\frac{3+2}{6}}+1$$

Add 3 and 2 to get 5.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=\frac{-\frac{1}{6}x}{\frac{5}{6}}+1$$

Divide -\frac{1}{6}x by \frac{5}{6} to get -\frac{1}{5}x.

$$\frac{\frac{x-5}{6}}{\frac{5}{6}}=-\frac{1}{5}x+1$$

Divide each term of x-5 by 6 to get \frac{1}{6}x-\frac{5}{6}.

$$\frac{\frac{1}{6}x-\frac{5}{6}}{\frac{5}{6}}=-\frac{1}{5}x+1$$

Divide each term of \frac{1}{6}x-\frac{5}{6} by \frac{5}{6} to get \frac{\frac{1}{6}x}{\frac{5}{6}}+\frac{-\frac{5}{6}}{\frac{5}{6}}.

$$\frac{\frac{1}{6}x}{\frac{5}{6}}+\frac{-\frac{5}{6}}{\frac{5}{6}}=-\frac{1}{5}x+1$$

Divide \frac{1}{6}x by \frac{5}{6} to get \frac{1}{5}x.

$$\frac{1}{5}x+\frac{-\frac{5}{6}}{\frac{5}{6}}=-\frac{1}{5}x+1$$

Divide -\frac{5}{6} by \frac{5}{6} to get -1.

$$\frac{1}{5}x-1=-\frac{1}{5}x+1$$

Add \frac{1}{5}x to both sides.

$$\frac{1}{5}x-1+\frac{1}{5}x=1$$

Combine \frac{1}{5}x and \frac{1}{5}x to get \frac{2}{5}x.

$$\frac{2}{5}x-1=1$$

Add 1 to both sides.

$$\frac{2}{5}x=1+1$$

Add 1 and 1 to get 2.

$$\frac{2}{5}x=2$$

Multiply both sides by \frac{5}{2}, the reciprocal of \frac{2}{5}.

$$x=2\times \frac{5}{2}$$

Cancel out 2 and 2.

$$x=5$$