Solve the Absolute Value Inequality for x |(x-1)/(3x-6)|=(1-x)/(3x-6)

$| \frac{x - 1}{3 x - 6} | = \frac{1 - x}{3 x - 6}$

Step 1

Multiply both sides by $3 x - 6$ .

$| \frac{x - 1}{3 x - 6} | (3 x - 6) = \frac{1 - x}{3 x - 6} (3 x - 6)$

Step 2

Simplify.

Simplify the left side.

Simplify $| \frac{x - 1}{3 x - 6} | (3 x - 6)$ .

Factor $3$ out of $3 x - 6$ .

Factor $3$ out of $3 x$ .

$| \frac{x - 1}{3 (x) - 6} | (3 x - 6) = \frac{1 - x}{3 x - 6} (3 x - 6)$

Factor $3$ out of $- 6$ .

$| \frac{x - 1}{3 x + 3 \cdot - 2} | (3 x - 6) = \frac{1 - x}{3 x - 6} (3 x - 6)$

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$| \frac{x - 1}{3 (x - 2)} | (3 x - 6) = \frac{1 - x}{3 x - 6} (3 x - 6)$

Remove non-negative terms from the absolute value.

$\frac{| x - 1 |}{3 | x - 2 |} (3 x - 6) = \frac{1 - x}{3 x - 6} (3 x - 6)$

Multiply $\frac{| x - 1 |}{3 | x - 2 |}$ by $3 x - 6$ .

$\frac{| x - 1 | (3 x - 6)}{3 | x - 2 |} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Cancel the common factor of $3 x - 6$ and $3$ .

Factor $3$ out of $| x - 1 | (3 x - 6)$ .

$\frac{3 (| x - 1 | (x - 2))}{3 | x - 2 |} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Cancel the common factors.

Factor $3$ out of $3 | x - 2 |$ .

$\frac{3 (| x - 1 | (x - 2))}{3 (| x - 2 |)} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Cancel the common factor.

$\frac{3 (| x - 1 | (x - 2))}{3 | x - 2 |} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Rewrite the expression.

$\frac{| x - 1 | (x - 2)}{| x - 2 |} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Simplify the right side.

Simplify $\frac{1 - x}{3 x - 6} (3 x - 6)$ .

Cancel the common factor of $3 x - 6$ .

Cancel the common factor.

$\frac{| x - 1 | (x - 2)}{| x - 2 |} = \frac{1 - x}{3 x - 6} (3 x - 6)$

Rewrite the expression.

$\frac{| x - 1 | (x - 2)}{| x - 2 |} = 1 - x$

Reorder $1$ and $- x$ .

$\frac{| x - 1 | (x - 2)}{| x - 2 |} = - x + 1$

Step 3

Solve for $x$ .

Multiply both sides by $| x - 2 |$ .

$\frac{| x - 1 | (x - 2)}{| x - 2 |} | x - 2 | = (- x + 1) | x - 2 |$

Simplify.

Simplify the left side.

Simplify $\frac{| x - 1 | (x - 2)}{| x - 2 |} | x - 2 |$ .

Cancel the common factor of $| x - 2 |$ .

Cancel the common factor.

$\frac{| x - 1 | (x - 2)}{| x - 2 |} | x - 2 | = (- x + 1) | x - 2 |$

Rewrite the expression.

$| x - 1 | (x - 2) = (- x + 1) | x - 2 |$

Apply the distributive property.

$| x - 1 | x + | x - 1 | \cdot - 2 = (- x + 1) | x - 2 |$

Simplify the expression.

Move $- 2$ to the left of $| x - 1 |$ .

$| x - 1 | x - 2 \cdot | x - 1 | = (- x + 1) | x - 2 |$

Reorder factors in $| x - 1 | x - 2 | x - 1 |$ .

$x | x - 1 | - 2 | x - 1 | = (- x + 1) | x - 2 |$

Simplify the right side.

Simplify $(- x + 1) | x - 2 |$ .

Apply the distributive property.

$x | x - 1 | - 2 | x - 1 | = - x | x - 2 | + 1 | x - 2 |$

Multiply $| x - 2 |$ by $1$ .

$x | x - 1 | - 2 | x - 1 | = - x | x - 2 | + | x - 2 |$

Solve for $x$ .

Rewrite the equation as $- x | x - 2 | + | x - 2 | = x | x - 1 | - 2 | x - 1 |$ .

$- x | x - 2 | + | x - 2 | = x | x - 1 | - 2 | x - 1 |$

Factor $| x - 2 |$ out of $- x | x - 2 | + | x - 2 |$ .

Factor $| x - 2 |$ out of $- x | x - 2 |$ .

$| x - 2 | (- x) + | x - 2 | = x | x - 1 | - 2 | x - 1 |$

Raise $| x - 2 |$ to the power of $1$ .

$| x - 2 | (- x) + | x - 2 | = x | x - 1 | - 2 | x - 1 |$

Factor $| x - 2 |$ out of ${| x - 2 |}^{1}$ .

$| x - 2 | (- x) + | x - 2 | \cdot 1 = x | x - 1 | - 2 | x - 1 |$

Factor $| x - 2 |$ out of $| x - 2 | (- x) + | x - 2 | \cdot 1$ .

$| x - 2 | (- x + 1) = x | x - 1 | - 2 | x - 1 |$

Divide each term in $| x - 2 | (- x + 1) = x | x - 1 | - 2 | x - 1 |$ by $- x + 1$ and simplify.

Divide each term in $| x - 2 | (- x + 1) = x | x - 1 | - 2 | x - 1 |$ by $- x + 1$ .

$\frac{| x - 2 | (- x + 1)}{- x + 1} = \frac{x | x - 1 |}{- x + 1} + \frac{- 2 | x - 1 |}{- x + 1}$

Simplify the left side.

Cancel the common factor of $- x + 1$ .

Cancel the common factor.

$\frac{| x - 2 | (- x + 1)}{- x + 1} = \frac{x | x - 1 |}{- x + 1} + \frac{- 2 | x - 1 |}{- x + 1}$

Divide $| x - 2 |$ by $1$ .

$| x - 2 | = \frac{x | x - 1 |}{- x + 1} + \frac{- 2 | x - 1 |}{- x + 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$| x - 2 | = \frac{x | x - 1 | - 2 | x - 1 |}{- x + 1}$

Factor $| x - 1 |$ out of $x | x - 1 | - 2 | x - 1 |$ .

Factor $| x - 1 |$ out of $x | x - 1 |$ .

$| x - 2 | = \frac{| x - 1 | x - 2 | x - 1 |}{- x + 1}$

Factor $| x - 1 |$ out of $- 2 | x - 1 |$ .

$| x - 2 | = \frac{| x - 1 | x + | x - 1 | \cdot - 2}{- x + 1}$

Factor $| x - 1 |$ out of $| x - 1 | x + | x - 1 | \cdot - 2$ .

$| x - 2 | = \frac{| x - 1 | (x - 2)}{- x + 1}$

Factor $- 1$ out of $- x$ .

$| x - 2 | = \frac{| x - 1 | (x - 2)}{- (x) + 1}$

Rewrite $1$ as $- 1 (- 1)$ .

$| x - 2 | = \frac{| x - 1 | (x - 2)}{- (x) - 1 (- 1)}$

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$| x - 2 | = \frac{| x - 1 | (x - 2)}{- (x - 1)}$

Rewrite negatives.

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$| x - 2 | = \frac{| x - 1 | (x - 2)}{- 1 (x - 1)}$

Move the negative in front of the fraction.

$| x - 2 | = - \frac{| x - 1 | (x - 2)}{x - 1}$

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

$x - 2 = - \frac{(x - 1) (x - 2)}{x - 1}$

$x - 2 = \frac{(x - 1) (x - 2)}{x - 1}$

$- (x - 2) = - \frac{(x - 1) (x - 2)}{x - 1}$

$- (x - 2) = \frac{(x - 1) (x - 2)}{x - 1}$

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

$x - 2 = - \frac{(x - 1) (x - 2)}{x - 1}$

$x - 2 = \frac{(x - 1) (x - 2)}{x - 1}$

Solve $x - 2 = - \frac{(x - 1) (x - 2)}{x - 1}$ for $x$ .

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

Add $x$ to both sides of the equation.

$x + x = 4$

Add $x$ and $x$ .

$2 x = 4$

Divide each term in $2 x = 4$ by $2$ and simplify.

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

Solve $x - 2 = \frac{(x - 1) (x - 2)}{x - 1}$ for $x$ .

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

Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

List all of the solutions.

$x = 2$

Step 4

Exclude the solutions that do not make $| \frac{x - 1}{3 x - 6} | = \frac{1 - x}{3 x - 6}$ true.

$No solution$

Solve the Absolute Value Inequality for x |(x-1)/(3x-6)|=(1-x)/(3x-6)

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