Pre-Algebra Examples

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

Step 1

To write $- \frac{5}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{5}{x - 1} \cdot \frac{x}{x} + \frac{2 - 3 x}{x}$

Step 2

To write $\frac{2 - 3 x}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$- \frac{5}{x - 1} \cdot \frac{x}{x} + \frac{2 - 3 x}{x} \cdot \frac{x - 1}{x - 1}$

Step 3

Write each expression with a common denominator of $(x - 1) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{x - 1}$ by $\frac{x}{x}$ .

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - 1) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(2 - 3 x) (x - 1)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $2$ by $- 1$ .

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

Step 5.2.1.2

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

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Step 5.2.1.2.1

Move $x$ .

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

Step 5.2.1.2.2

Multiply $x$ by $x$ .

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

Step 5.2.1.3

Multiply $- 1$ by $- 3$ .

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

Step 5.2.2

Add $2 x$ and $3 x$ .

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

Step 5.3

Add $- 5 x$ and $5 x$ .

$\frac{0 - 2 - 3 x^{2}}{x (x - 1)}$

Step 5.4

Subtract $2$ from $0$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 6.3

Factor $- 1$ out of $- 1 (2) - (3 x^{2})$ .

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

Step 6.4

Move the negative in front of the fraction.

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