Solve for x |x-3|=|9-2x|

$| x - 3 | = | 9 - 2 x |$

Step 1

Rewrite the equation as $| 9 - 2 x | = | x - 3 |$ .

$| 9 - 2 x | = | x - 3 |$

Step 2

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

$9 - 2 x = x - 3$

$9 - 2 x = - (x - 3)$

$- (9 - 2 x) = x - 3$

$- (9 - 2 x) = - (x - 3)$

Step 3

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

$9 - 2 x = x - 3$

$9 - 2 x = - (x - 3)$

Step 4

Solve $9 - 2 x = x - 3$ for $x$ .

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

Subtract $x$ from both sides of the equation.

$9 - 2 x - x = - 3$

Subtract $x$ from $- 2 x$ .

$9 - 3 x = - 3$

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

Subtract $9$ from both sides of the equation.

$- 3 x = - 3 - 9$

Subtract $9$ from $- 3$ .

$- 3 x = - 12$

Divide each term in $- 3 x = - 12$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 12$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 12}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 12}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 12}{- 3}$

Simplify the right side.

Divide $- 12$ by $- 3$ .

$x = 4$

Step 5

Solve $9 - 2 x = - (x - 3)$ for $x$ .

Simplify $- (x - 3)$ .

Rewrite.

$9 - 2 x = 0 + 0 - (x - 3)$

Simplify by adding zeros.

$9 - 2 x = - (x - 3)$

Apply the distributive property.

$9 - 2 x = - x - - 3$

Multiply $- 1$ by $- 3$ .

$9 - 2 x = - x + 3$

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

Add $x$ to both sides of the equation.

$9 - 2 x + x = 3$

Add $- 2 x$ and $x$ .

$9 - x = 3$

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

Subtract $9$ from both sides of the equation.

$- x = 3 - 9$

Subtract $9$ from $3$ .

$- x = - 6$

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

Step 6

List all of the solutions.

$x = 4, 6$

Solve for x |x-3|=|9-2x|

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