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

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

Solve for x

$$x = \frac{29}{4} = 7\frac{1}{4} = 7.25$$

Steps for Solving Linear Equation

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

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

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

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

To find the opposite of 1+x, find the opposite of each term.

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

Subtract 1 from -3 to get -4.

$$\frac{3}{2}x-4-x+\frac{1}{3}\left(2x+\frac{1}{2}\right)=\frac{1}{2}x+1$$

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

$$\frac{1}{2}x-4+\frac{1}{3}\left(2x+\frac{1}{2}\right)=\frac{1}{2}x+1$$

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

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

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

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

Multiply \frac{1}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 1}{3\times 2}.

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

Combine \frac{1}{2}x and \frac{2}{3}x to get \frac{7}{6}x.

$$\frac{7}{6}x-4+\frac{1}{6}=\frac{1}{2}x+1$$

Convert -4 to fraction -\frac{24}{6}.

$$\frac{7}{6}x-\frac{24}{6}+\frac{1}{6}=\frac{1}{2}x+1$$

Since -\frac{24}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{7}{6}x+\frac{-24+1}{6}=\frac{1}{2}x+1$$

Add -24 and 1 to get -23.

$$\frac{7}{6}x-\frac{23}{6}=\frac{1}{2}x+1$$

Subtract \frac{1}{2}x from both sides.

$$\frac{7}{6}x-\frac{23}{6}-\frac{1}{2}x=1$$

Combine \frac{7}{6}x and -\frac{1}{2}x to get \frac{2}{3}x.

$$\frac{2}{3}x-\frac{23}{6}=1$$

Add \frac{23}{6} to both sides.

$$\frac{2}{3}x=1+\frac{23}{6}$$

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

$$\frac{2}{3}x=\frac{6}{6}+\frac{23}{6}$$

Since \frac{6}{6} and \frac{23}{6} have the same denominator, add them by adding their numerators.

$$\frac{2}{3}x=\frac{6+23}{6}$$

Add 6 and 23 to get 29.

$$\frac{2}{3}x=\frac{29}{6}$$

Multiply both sides by \frac{3}{2}, the reciprocal of \frac{2}{3}.

$$x=\frac{29}{6}\times \frac{3}{2}$$

Multiply \frac{29}{6} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{29\times 3}{6\times 2}$$

Do the multiplications in the fraction \frac{29\times 3}{6\times 2}.

$$x=\frac{87}{12}$$

Reduce the fraction \frac{87}{12} to lowest terms by extracting and canceling out 3.

$$x=\frac{29}{4}$$