Algebra Examples

$\frac{x}{2 x - 4} - \frac{5}{x + 1}$

Step 1

Factor $2$ out of $2 x - 4$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{x}{2 (x) - 4} - \frac{5}{x + 1}$

Step 1.2

Factor $2$ out of $- 4$ .

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

Step 1.3

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

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

Step 2

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

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

Step 3

To write $- \frac{5}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{2 (x - 2)}{2 (x - 2)}$ .

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

Step 4

Write each expression with a common denominator of $2 (x - 2) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{2 (x - 2)}$ by $\frac{x + 1}{x + 1}$ .

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

Step 4.2

Multiply $\frac{5}{x + 1}$ by $\frac{2 (x - 2)}{2 (x - 2)}$ .

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

Step 4.3

Reorder the factors of $2 (x - 2) (x + 1)$ .

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

Step 4.4

Reorder the factors of $(x + 1) (2 (x - 2))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $x$ by $1$ .

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

Step 6.4

Multiply $- 5$ by $2$ .

$\frac{x^{2} + x - 10 (x - 2)}{2 (x + 1) (x - 2)}$

Step 6.5

Apply the distributive property.

$\frac{x^{2} + x - 10 x - 10 \cdot - 2}{2 (x + 1) (x - 2)}$

Step 6.6

Multiply $- 10$ by $- 2$ .

$\frac{x^{2} + x - 10 x + 20}{2 (x + 1) (x - 2)}$

Step 6.7

Subtract $10 x$ from $x$ .

$\frac{x^{2} - 9 x + 20}{2 (x + 1) (x - 2)}$

Step 6.8

Factor $x^{2} - 9 x + 20$ using the AC method.

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Step 6.8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $- 9$ .

$- 5, - 4$

Step 6.8.2

Write the factored form using these integers.

$\frac{(x - 5) (x - 4)}{2 (x + 1) (x - 2)}$