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

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

Solve for x

$$x=23$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Add -1 and 4 to get 3.

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

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

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

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

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

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

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

Convert 2 to fraction \frac{6}{3}.

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

Since -\frac{1}{3} and \frac{6}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 6 from -1 to get -7.

$$\frac{1}{2}x+\frac{3}{2}=\frac{2}{3}x-\frac{7}{3}$$

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

$$\frac{1}{2}x+\frac{3}{2}-\frac{2}{3}x=-\frac{7}{3}$$

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

$$-\frac{1}{6}x+\frac{3}{2}=-\frac{7}{3}$$

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

$$-\frac{1}{6}x=-\frac{7}{3}-\frac{3}{2}$$

Least common multiple of 3 and 2 is 6. Convert -\frac{7}{3} and \frac{3}{2} to fractions with denominator 6.

$$-\frac{1}{6}x=-\frac{14}{6}-\frac{9}{6}$$

Since -\frac{14}{6} and \frac{9}{6} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{6}x=\frac{-14-9}{6}$$

Subtract 9 from -14 to get -23.

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

Multiply both sides by -6, the reciprocal of -\frac{1}{6}.

$$x=-\frac{23}{6}\left(-6\right)$$

Express -\frac{23}{6}\left(-6\right) as a single fraction.

$$x=\frac{-23\left(-6\right)}{6}$$

Multiply -23 and -6 to get 138.

$$x=\frac{138}{6}$$

Divide 138 by 6 to get 23.

$$x=23$$