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

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

Step 1

Rewrite the equation as $| x + 2 | = | x - 3 |$ .

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

Step 2

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

$x + 2 = x - 3$

$x + 2 = - (x - 3)$

$- (x + 2) = x - 3$

$- (x + 2) = - (x - 3)$

Step 3

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

$x + 2 = x - 3$

$x + 2 = - (x - 3)$

Step 4

Solve $x + 2 = x - 3$ for $x$ .

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

Subtract $x$ from both sides of the equation.

$x + 2 - x = - 3$

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

Subtract $x$ from $x$ .

$0 + 2 = - 3$

Add $0$ and $2$ .

$2 = - 3$

Since $2 \neq - 3$ , there are no solutions.

No solution

Step 5

Solve $x + 2 = - (x - 3)$ for $x$ .

Simplify $- (x - 3)$ .

Rewrite.

$x + 2 = 0 + 0 - (x - 3)$

Simplify by adding zeros.

$x + 2 = - (x - 3)$

Apply the distributive property.

$x + 2 = - x - - 3$

Multiply $- 1$ by $- 3$ .

$x + 2 = - x + 3$

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

Add $x$ to both sides of the equation.

$x + 2 + x = 3$

Add $x$ and $x$ .

$2 x + 2 = 3$

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

Subtract $2$ from both sides of the equation.

$2 x = 3 - 2$

Subtract $2$ from $3$ .

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6

List all of the solutions.

$x = \frac{1}{2}$

Step 7

Verify each of the solutions by substituting them into $| x - 3 | = | x + 2 |$ and solving.

$x = \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$

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

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