Solve 3(x+1)-1/3(x+1)=2(x-1)-1/2(x+1)

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

Solve for x

$$x = -\frac{31}{7} = -4\frac{3}{7} \approx -4.428571429$$

Steps for Solving Linear Equation

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

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

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

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

Combine 3x and -\frac{1}{3}x to get \frac{8}{3}x.

$$\frac{8}{3}x+3-\frac{1}{3}=2\left(x-1\right)-\frac{1}{2}\left(x+1\right)$$

Convert 3 to fraction \frac{9}{3}.

$$\frac{8}{3}x+\frac{9}{3}-\frac{1}{3}=2\left(x-1\right)-\frac{1}{2}\left(x+1\right)$$

Since \frac{9}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8}{3}x+\frac{9-1}{3}=2\left(x-1\right)-\frac{1}{2}\left(x+1\right)$$

Subtract 1 from 9 to get 8.

$$\frac{8}{3}x+\frac{8}{3}=2\left(x-1\right)-\frac{1}{2}\left(x+1\right)$$

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

$$\frac{8}{3}x+\frac{8}{3}=2x-2-\frac{1}{2}\left(x+1\right)$$

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

$$\frac{8}{3}x+\frac{8}{3}=2x-2-\frac{1}{2}x-\frac{1}{2}$$

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

$$\frac{8}{3}x+\frac{8}{3}=\frac{3}{2}x-2-\frac{1}{2}$$

Convert -2 to fraction -\frac{4}{2}.

$$\frac{8}{3}x+\frac{8}{3}=\frac{3}{2}x-\frac{4}{2}-\frac{1}{2}$$

Since -\frac{4}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8}{3}x+\frac{8}{3}=\frac{3}{2}x+\frac{-4-1}{2}$$

Subtract 1 from -4 to get -5.

$$\frac{8}{3}x+\frac{8}{3}=\frac{3}{2}x-\frac{5}{2}$$

Subtract \frac{3}{2}x from both sides.

$$\frac{8}{3}x+\frac{8}{3}-\frac{3}{2}x=-\frac{5}{2}$$

Combine \frac{8}{3}x and -\frac{3}{2}x to get \frac{7}{6}x.

$$\frac{7}{6}x+\frac{8}{3}=-\frac{5}{2}$$

Subtract \frac{8}{3} from both sides.

$$\frac{7}{6}x=-\frac{5}{2}-\frac{8}{3}$$

Least common multiple of 2 and 3 is 6. Convert -\frac{5}{2} and \frac{8}{3} to fractions with denominator 6.

$$\frac{7}{6}x=-\frac{15}{6}-\frac{16}{6}$$

Since -\frac{15}{6} and \frac{16}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7}{6}x=\frac{-15-16}{6}$$

Subtract 16 from -15 to get -31.

$$\frac{7}{6}x=-\frac{31}{6}$$

Multiply both sides by \frac{6}{7}, the reciprocal of \frac{7}{6}.

$$x=-\frac{31}{6}\times \frac{6}{7}$$

Multiply -\frac{31}{6} times \frac{6}{7} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-31\times 6}{6\times 7}$$

Cancel out 6 in both numerator and denominator.

$$x=\frac{-31}{7}$$

Fraction \frac{-31}{7} can be rewritten as -\frac{31}{7} by extracting the negative sign.

$$x=-\frac{31}{7}$$