Solve for x 2|x+1|-x=3x-4

$2 | x + 1 | - x = 3 x - 4$

Step 1

Move all terms not containing $| x + 1 |$ to the right side of the equation.

Add $x$ to both sides of the equation.

$2 | x + 1 | = 3 x - 4 + x$

Add $3 x$ and $x$ .

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

Step 2

Divide each term in $2 | x + 1 | = 4 x - 4$ by $2$ and simplify.

Divide each term in $2 | x + 1 | = 4 x - 4$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $| x + 1 |$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 x$ .

$| x + 1 | = \frac{2 (2 x)}{2} + \frac{- 4}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$| x + 1 | = \frac{2 (2 x)}{2 (1)} + \frac{- 4}{2}$

Cancel the common factor.

$| x + 1 | = \frac{2 (2 x)}{2 \cdot 1} + \frac{- 4}{2}$

Rewrite the expression.

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

Divide $2 x$ by $1$ .

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

Divide $- 4$ by $2$ .

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

Step 3

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

$x + 1 = \pm (2 x - 2)$

Step 4

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.

$x + 1 = 2 x - 2$

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

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

$x + 1 - 2 x = - 2$

Subtract $2 x$ from $x$ .

$- x + 1 = - 2$

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

Subtract $1$ from both sides of the equation.

$- x = - 2 - 1$

Subtract $1$ from $- 2$ .

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

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

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

Simplify $- (2 x - 2)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply.

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 2$ .

$x + 1 = - 2 x + 2$

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

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

$x + 1 + 2 x = 2$

Add $x$ and $2 x$ .

$3 x + 1 = 2$

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

Subtract $1$ from both sides of the equation.

$3 x = 2 - 1$

Subtract $1$ from $2$ .

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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

$x = 3, \frac{1}{3}$

Step 5

Exclude the solutions that do not make $2 | x + 1 | - x = 3 x - 4$ true.

$x = 3$

Solve for x 2|x+1|-x=3x-4

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