Algebra Examples

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

Step 1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2 - x}{2 - x}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Factor $x$ out of $x (2 - x) + x$ .

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Step 3.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.3

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

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

Step 3.2

Add $2$ and $1$ .

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

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{x - 2}{x - 3}$ by $\frac{x (- x + 3)}{2 - x}$ .

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

Step 4.2

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

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Step 4.2.1

Factor $- 1$ out of $x$ .

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

Step 4.2.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 4.2.3

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

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

Step 4.2.4

Reorder terms.

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

Step 4.2.5

Cancel the common factor.

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

Step 4.2.6

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 4.3.3

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

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

Step 4.3.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 4.3.5

Cancel the common factor.

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

Step 4.3.6

Divide $- 1 x \cdot (- 1)$ by $1$ .

$- 1 x \cdot (- 1)$

Step 4.4

Rewrite $- 1 x$ as $- x$ .

$- x \cdot - 1$

Step 5

Multiply $- x \cdot - 1$ .

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Step 5.1

Multiply $- 1$ by $- 1$ .

$1 x$

Step 5.2

Multiply $x$ by $1$ .

$x$