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

$| x - 3 | = | 3 - x |$

Step 1

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

$| 3 - x | = | x - 3 |$

Step 2

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

$3 - x = x - 3$

$3 - x = - (x - 3)$

$- (3 - x) = x - 3$

$- (3 - x) = - (x - 3)$

Step 3

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

$3 - x = x - 3$

$3 - x = - (x - 3)$

Step 4

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

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

Subtract $x$ from both sides of the equation.

$3 - x - x = - 3$

Subtract $x$ from $- x$ .

$3 - 2 x = - 3$

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

Subtract $3$ from both sides of the equation.

$- 2 x = - 3 - 3$

Subtract $3$ from $- 3$ .

$- 2 x = - 6$

Divide each term in $- 2 x = - 6$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 6$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 6}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 2}$

Simplify the right side.

Divide $- 6$ by $- 2$ .

$x = 3$

Step 5

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

Simplify $- (x - 3)$ .

Rewrite.

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

Simplify by adding zeros.

$3 - x = - (x - 3)$

Apply the distributive property.

$3 - x = - x - - 3$

Multiply $- 1$ by $- 3$ .

$3 - x = - x + 3$

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

Add $x$ to both sides of the equation.

$3 - x + x = 3$

Combine the opposite terms in $3 - x + x$ .

Add $- x$ and $x$ .

$3 + 0 = 3$

Add $3$ and $0$ .

$3 = 3$

Since $3 = 3$ , the equation will always be true.

Always true

Step 6

List all of the solutions.

$x = 3$

Step 7

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

$x = 3$

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

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