Solve the Absolute Value Inequality for x |x|+x=0

$| x | + x = 0$

Step 1

Subtract $x$ from both sides of the equation.

$| x | = - x$

Step 2

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

$x = \pm (- x)$

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.

$x = - x$

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

Add $x$ to both sides of the equation.

$x + x = 0$

Add $x$ and $x$ .

$2 x = 0$

Divide each term in $2 x = 0$ by $2$ and simplify.

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x = 0$

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

$x = x$

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

Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

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

$x = 0,$

Step 4

Verify each of the solutions by substituting them into $| x | + x = 0$ and solving.

$x = 0$