Algebra Examples

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

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.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

Factor $3$ out of $3 x$ .

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

Step 2.1.1.1.2

Factor $3$ out of $- 6$ .

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

Step 2.1.1.1.3

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

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

Step 2.1.1.2

Remove non-negative terms from the absolute value.

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

Cancel the common factors.

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Step 2.1.1.4.2.1

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

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

Step 2.1.1.4.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.3

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

Rewrite the expression.

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

Step 2.2.1.2

Reorder $1$ and $- x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

Cancel the common factor.

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

Step 3.2.1.1.1.2

Rewrite the expression.

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

Step 3.2.1.1.2

Apply the distributive property.

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

Step 3.2.1.1.3

Simplify the expression.

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Step 3.2.1.1.3.1

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

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

Step 3.2.1.1.3.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

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

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

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.2.4

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

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

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

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

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

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Combine the numerators over the common denominator.

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

Step 3.3.3.3.2

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

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Step 3.3.3.3.2.1

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

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

Step 3.3.3.3.2.2

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

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

Step 3.3.3.3.2.3

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

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

Step 3.3.3.3.3

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

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

Step 3.3.3.3.4

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

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

Step 3.3.3.3.5

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

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

Step 3.3.3.3.6

Rewrite negatives.

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Step 3.3.3.3.6.1

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

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

Step 3.3.3.3.6.2

Move the negative in front of the fraction.

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

Step 3.3.4

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

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

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

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

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

Step 3.3.5

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

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

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

Step 3.3.6

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

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Step 3.3.6.1

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

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Step 3.3.6.1.1

Add $x$ to both sides of the equation.

$x + x = 4$

Step 3.3.6.1.2

Add $x$ and $x$ .

$2 x = 4$

Step 3.3.6.2

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

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Step 3.3.6.2.1

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

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

Step 3.3.6.2.2

Simplify the left side.

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Step 3.3.6.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.6.2.2.1.1

Cancel the common factor.

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

Step 3.3.6.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.6.2.3

Simplify the right side.

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Step 3.3.6.2.3.1

Divide $4$ by $2$ .

$x = 2$

Step 3.3.7

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

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Step 3.3.7.1

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

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Step 3.3.7.1.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.3.7.1.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.3.7.2

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

Always true

Step 3.3.8

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.

$No solution$