Solve for x |2x-9|=|x-6|

$| 2 x - 9 | = | x - 6 |$

Rewrite the equation as $| x - 6 | = | 2 x - 9 |$ .

$| x - 6 | = | 2 x - 9 |$

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

$x - 6 = 2 x - 9$

$x - 6 = - (2 x - 9)$

$- (x - 6) = 2 x - 9$

$- (x - 6) = - (2 x - 9)$

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

$x - 6 = 2 x - 9$

$x - 6 = - (2 x - 9)$

Solve $x - 6 = 2 x - 9$ for $x$ .

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

Subtract $2 x$ from both sides of the equation.

$x - 6 - 2 x = - 9$

Subtract $2 x$ from $x$ .

$- x - 6 = - 9$

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

Add $6$ to both sides of the equation.

$- x = - 9 + 6$

Add $- 9$ and $6$ .

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

Solve $x - 6 = - (2 x - 9)$ for $x$ .

Simplify $- (2 x - 9)$ .

Rewrite.

$x - 6 = 0 + 0 - (2 x - 9)$

Simplify by adding zeros.

$x - 6 = - (2 x - 9)$

Apply the distributive property.

$x - 6 = - (2 x) - - 9$

Multiply.

Multiply $2$ by $- 1$ .

$x - 6 = - 2 x - - 9$

Multiply $- 1$ by $- 9$ .

$x - 6 = - 2 x + 9$

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

Add $2 x$ to both sides of the equation.

$x - 6 + 2 x = 9$

Add $x$ and $2 x$ .

$3 x - 6 = 9$

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

Add $6$ to both sides of the equation.

$3 x = 9 + 6$

Add $9$ and $6$ .

$3 x = 15$

Divide each term in $3 x = 15$ by $3$ and simplify.

Divide each term in $3 x = 15$ by $3$ .

$\frac{3 x}{3} = \frac{15}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{15}{3}$

Divide $x$ by $1$ .

$x = \frac{15}{3}$

Simplify the right side.

Divide $15$ by $3$ .

$x = 5$

List all of the solutions.

$x = 3, 5$

Solve for x |2x-9|=|x-6|

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