Algebra Examples

$\frac{\frac{3}{x - 1} - 4}{2 - \frac{2}{x - 1}}$

Step 1

Multiply the numerator and denominator of the fraction by $x - 1$ .

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Step 1.1

Multiply $\frac{\frac{3}{x - 1} - 4}{2 - \frac{2}{x - 1}}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x - 1}{x - 1} \cdot \frac{\frac{3}{x - 1} - 4}{2 - \frac{2}{x - 1}}$

Step 1.2

Combine.

$\frac{(x - 1) (\frac{3}{x - 1} - 4)}{(x - 1) (2 - \frac{2}{x - 1})}$

Step 2

Apply the distributive property.

$\frac{(x - 1) \frac{3}{x - 1} + (x - 1) \cdot - 4}{(x - 1) \cdot 2 + (x - 1) (- \frac{2}{x - 1})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 1$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x - 1) \frac{3}{x - 1} + (x - 1) \cdot - 4}{(x - 1) \cdot 2 + (x - 1) (- \frac{2}{x - 1})}$

Step 3.1.2

Rewrite the expression.

$\frac{3 + (x - 1) \cdot - 4}{(x - 1) \cdot 2 + (x - 1) (- \frac{2}{x - 1})}$

Step 3.2

Cancel the common factor of $x - 1$ .

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Step 3.2.1

Move the leading negative in $- \frac{2}{x - 1}$ into the numerator.

$\frac{3 + (x - 1) \cdot - 4}{(x - 1) \cdot 2 + (x - 1) \frac{- 2}{x - 1}}$

Step 3.2.2

Cancel the common factor.

$\frac{3 + (x - 1) \cdot - 4}{(x - 1) \cdot 2 + (x - 1) \frac{- 2}{x - 1}}$

Step 3.2.3

Rewrite the expression.

$\frac{3 + (x - 1) \cdot - 4}{(x - 1) \cdot 2 - 2}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 + x \cdot - 4 - 1 \cdot - 4}{(x - 1) \cdot 2 - 2}$

Step 4.2

Move $- 4$ to the left of $x$ .

$\frac{3 - 4 \cdot x - 1 \cdot - 4}{(x - 1) \cdot 2 - 2}$

Step 4.3

Multiply $- 1$ by $- 4$ .

$\frac{3 - 4 \cdot x + 4}{(x - 1) \cdot 2 - 2}$

Step 4.4

Add $3$ and $4$ .

$\frac{- 4 x + 7}{(x - 1) \cdot 2 - 2}$

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{- 4 x + 7}{2 \cdot (x - 1) - 2}$

Step 5.1.2

Factor $2$ out of $- 2$ .

$\frac{- 4 x + 7}{2 \cdot (x - 1) + 2 (- 1)}$

Step 5.1.3

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

$\frac{- 4 x + 7}{2 (x - 1 - 1)}$

Step 5.2

Subtract $1$ from $- 1$ .

$\frac{- 4 x + 7}{2 (x - 2)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) + 7}{2 (x - 2)}$

Step 6.2

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (4 x) - 1 (- 7)}{2 (x - 2)}$

Step 6.3

Factor $- 1$ out of $- (4 x) - 1 (- 7)$ .

$\frac{- (4 x - 7)}{2 (x - 2)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (4 x - 7)$ as $- 1 (4 x - 7)$ .

$\frac{- 1 (4 x - 7)}{2 (x - 2)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{4 x - 7}{2 (x - 2)}$