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

$| 9 x + 6 | = | x - 9 |$

Step 1

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

$| x - 9 | = | 9 x + 6 |$

Step 2

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

$x - 9 = 9 x + 6$

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

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

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

Step 3

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

$x - 9 = 9 x + 6$

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

Step 4

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

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

Subtract $9 x$ from both sides of the equation.

$x - 9 - 9 x = 6$

Subtract $9 x$ from $x$ .

$- 8 x - 9 = 6$

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

Add $9$ to both sides of the equation.

$- 8 x = 6 + 9$

Add $6$ and $9$ .

$- 8 x = 15$

Divide each term in $- 8 x = 15$ by $- 8$ and simplify.

Divide each term in $- 8 x = 15$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{15}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{15}{- 8}$

Divide $x$ by $1$ .

$x = \frac{15}{- 8}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{15}{8}$

Step 5

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

Simplify $- (9 x + 6)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

$x - 9 = - (9 x) - 1 \cdot 6$

Multiply.

Multiply $9$ by $- 1$ .

$x - 9 = - 9 x - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$x - 9 = - 9 x - 6$

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

Add $9 x$ to both sides of the equation.

$x - 9 + 9 x = - 6$

Add $x$ and $9 x$ .

$10 x - 9 = - 6$

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

Add $9$ to both sides of the equation.

$10 x = - 6 + 9$

Add $- 6$ and $9$ .

$10 x = 3$

Divide each term in $10 x = 3$ by $10$ and simplify.

Divide each term in $10 x = 3$ by $10$ .

$\frac{10 x}{10} = \frac{3}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 x}{10} = \frac{3}{10}$

Divide $x$ by $1$ .

$x = \frac{3}{10}$

Step 6

List all of the solutions.

$x = - \frac{15}{8}, \frac{3}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15}{8}, \frac{3}{10}$

Decimal Form:

$x = - 1.875, 0.3$

Mixed Number Form:

$x = - 1 \frac{7}{8}, \frac{3}{10}$

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

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