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

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

Solve for x

$$x = -\frac{31}{12} = -2\frac{7}{12} \approx -2.583333333$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6, the least common multiple of 2,3.

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

Use the distributive property to multiply 3 by x+1.

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

Use the distributive property to multiply -2 by x-1.

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

Combine 3x and -2x to get x.

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

Add 3 and 2 to get 5.

$$x+5+6=3\left(\frac{1}{3}x-\frac{3}{2}\right)-6x$$

Add 5 and 6 to get 11.

$$x+11=3\left(\frac{1}{3}x-\frac{3}{2}\right)-6x$$

Use the distributive property to multiply 3 by \frac{1}{3}x-\frac{3}{2}.

$$x+11=3\times \frac{1}{3}x+3\left(-\frac{3}{2}\right)-6x$$

Cancel out 3 and 3.

$$x+11=x+3\left(-\frac{3}{2}\right)-6x$$

Express 3\left(-\frac{3}{2}\right) as a single fraction.

$$x+11=x+\frac{3\left(-3\right)}{2}-6x$$

Multiply 3 and -3 to get -9.

$$x+11=x+\frac{-9}{2}-6x$$

Fraction \frac{-9}{2} can be rewritten as -\frac{9}{2} by extracting the negative sign.

$$x+11=x-\frac{9}{2}-6x$$

Combine x and -6x to get -5x.

$$x+11=-5x-\frac{9}{2}$$

Add 5x to both sides.

$$x+11+5x=-\frac{9}{2}$$

Combine x and 5x to get 6x.

$$6x+11=-\frac{9}{2}$$

Subtract 11 from both sides.

$$6x=-\frac{9}{2}-11$$

Convert 11 to fraction \frac{22}{2}.

$$6x=-\frac{9}{2}-\frac{22}{2}$$

Since -\frac{9}{2} and \frac{22}{2} have the same denominator, subtract them by subtracting their numerators.

$$6x=\frac{-9-22}{2}$$

Subtract 22 from -9 to get -31.

$$6x=-\frac{31}{2}$$

Divide both sides by 6.

$$x=\frac{-\frac{31}{2}}{6}$$

Express \frac{-\frac{31}{2}}{6} as a single fraction.

$$x=\frac{-31}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$x=\frac{-31}{12}$$

Fraction \frac{-31}{12} can be rewritten as -\frac{31}{12} by extracting the negative sign.

$$x=-\frac{31}{12}$$