Solve the Absolute Value Inequality for x |2x+3|=x

$| 2 x + 3 | = x$

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

$2 x + 3 = \pm x$

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.

$2 x + 3 = x$

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

Subtract $x$ from both sides of the equation.

$2 x + 3 - x = 0$

Subtract $x$ from $2 x$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$2 x + 3 = - x$

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

Add $x$ to both sides of the equation.

$2 x + 3 + x = 0$

Add $2 x$ and $x$ .

$3 x + 3 = 0$

Subtract $3$ from both sides of the equation.

$3 x = - 3$

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 $3$ .

$x = - 1$

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

$x = - 3, - 1$

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

$No solution$