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

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

Step 1

Subtract $x$ from both sides of the equation.

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

Step 2

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

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

Step 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.

$x + 2 = - 2 - x$

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

Add $x$ to both sides of the equation.

$x + 2 + x = - 2$

Add $x$ and $x$ .

$2 x + 2 = - 2$

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

Subtract $2$ from both sides of the equation.

$2 x = - 2 - 2$

Subtract $2$ from $- 2$ .

$2 x = - 4$

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.

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

Simplify $- (- 2 - x)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

$x + 2 = 2 - - x$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$x + 2 = 2 + 1 x$

Multiply $x$ by $1$ .

$x + 2 = 2 + x$

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

Subtract $x$ from both sides of the equation.

$x + 2 - x = 2$

Combine the opposite terms in $x + 2 - x$ .

Subtract $x$ from $x$ .

$0 + 2 = 2$

Add $0$ and $2$ .

$2 = 2$

Since $2 = 2$ , the equation will always be true.

Always true

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

$x + 2 = - 2,$

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

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