Algebra Examples

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

Step 1

Find the common denominator.

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Step 1.1

Write $2$ as a fraction with denominator $1$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Write $- x$ as a fraction with denominator $1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $2$ by $3$ .

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

Step 3.3

Multiply $- 1$ by $2$ .

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Multiply $3$ by $- 1$ .

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

Step 3.6

Rewrite using the commutative property of multiplication.

$\frac{6 - 2 x - 3 x - 1 \cdot - 1 x \cdot x - 1}{- x + 3}$

Step 3.7

Simplify each term.

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Step 3.7.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.7.1.1

Move $x$ .

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

Step 3.7.1.2

Multiply $x$ by $x$ .

$\frac{6 - 2 x - 3 x - 1 \cdot - 1 x^{2} - 1}{- x + 3}$

Step 3.7.2

Multiply $- 1$ by $- 1$ .

$\frac{6 - 2 x - 3 x + 1 x^{2} - 1}{- x + 3}$

Step 3.7.3

Multiply $x^{2}$ by $1$ .

$\frac{6 - 2 x - 3 x + x^{2} - 1}{- x + 3}$

Step 4

Simplify terms.

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Step 4.1

Subtract $1$ from $6$ .

$\frac{- 2 x - 3 x + x^{2} + 5}{- x + 3}$

Step 4.2

Subtract $3 x$ from $- 2 x$ .

$\frac{- 5 x + x^{2} + 5}{- x + 3}$

Step 4.3

Reorder $- 5 x$ and $x^{2}$ .

$\frac{x^{2} - 5 x + 5}{- x + 3}$

Step 4.4

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

$\frac{x^{2} - 5 x + 5}{- (x) + 3}$

Step 4.5

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

$\frac{x^{2} - 5 x + 5}{- (x) - 1 (- 3)}$

Step 4.6

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

$\frac{x^{2} - 5 x + 5}{- (x - 3)}$

Step 4.7

Rewrite negatives.

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Step 4.7.1

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

$\frac{x^{2} - 5 x + 5}{- 1 (x - 3)}$

Step 4.7.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 x + 5}{x - 3}$