Find All Complex Solutions |2x-1|=3

$| 2 x - 1 | = 3$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 x - 1 = \pm 3$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$2 x - 1 = 3$

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$2 x = 3 + 1$

Add $3$ and $1$ .

$2 x = 4$

Divide each term in $2 x = 4$ by $2$ and simplify.

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$2 x - 1 = - 3$

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$2 x = - 3 + 1$

Add $- 3$ and $1$ .

$2 x = - 2$

Divide each term in $2 x = - 2$ by $2$ and simplify.

Divide each term in $2 x = - 2$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $2$ .

$x = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 1$