Find All Complex Solutions |2x-4|=|x+1|

$| 2 x - 4 | = | x + 1 |$

Step 1

Rewrite the equation as $| x + 1 | = | 2 x - 4 |$ .

$| x + 1 | = | 2 x - 4 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$x + 1 = 2 x - 4$

$x + 1 = - (2 x - 4)$

$- (x + 1) = 2 x - 4$

$- (x + 1) = - (2 x - 4)$

Step 3

After simplifying, there are only two unique equations to be solved.

$x + 1 = 2 x - 4$

$x + 1 = - (2 x - 4)$

Step 4

Solve $x + 1 = 2 x - 4$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$x + 1 - 2 x = - 4$

Subtract $2 x$ from $x$ .

$- x + 1 = - 4$

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

Subtract $1$ from both sides of the equation.

$- x = - 4 - 1$

Subtract $1$ from $- 4$ .

$- x = - 5$

Divide each term in $- x = - 5$ by $- 1$ and simplify.

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x = 5$

Step 5

Solve $x + 1 = - (2 x - 4)$ for $x$ .

Simplify $- (2 x - 4)$ .

Rewrite.

$x + 1 = 0 + 0 - (2 x - 4)$

Simplify by adding zeros.

$x + 1 = - (2 x - 4)$

Apply the distributive property.

$x + 1 = - (2 x) - - 4$

Multiply.

Multiply $2$ by $- 1$ .

$x + 1 = - 2 x - - 4$

Multiply $- 1$ by $- 4$ .

$x + 1 = - 2 x + 4$

Move all terms containing $x$ to the left side of the equation.

Add $2 x$ to both sides of the equation.

$x + 1 + 2 x = 4$

Add $x$ and $2 x$ .

$3 x + 1 = 4$

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

Subtract $1$ from both sides of the equation.

$3 x = 4 - 1$

Subtract $1$ from $4$ .

$3 x = 3$

Divide each term in $3 x = 3$ by $3$ and simplify.

Divide each term in $3 x = 3$ by $3$ .

$\frac{3 x}{3} = \frac{3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{3}{3}$

Divide $x$ by $1$ .

$x = \frac{3}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$x = 1$

Step 6

List all of the solutions.

$x = 5, 1$

Find All Complex Solutions |2x-4|=|x+1|

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