Algebra Examples

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

Step 1

Factor $3$ out of $3 x + 3$ .

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Step 1.1

Factor $3$ out of $3 x$ .

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

Step 1.2

Factor $3$ out of $3$ .

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

Step 1.3

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

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

Step 2

To write $\frac{2 x}{3 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Step 3

To write $- \frac{2}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{3 (x + 1)}{3 (x + 1)}$ .

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

Step 4

Write each expression with a common denominator of $3 (x + 1) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 x}{3 (x + 1)}$ by $\frac{x + 5}{x + 5}$ .

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

Step 4.2

Multiply $\frac{2}{x + 5}$ by $\frac{3 (x + 1)}{3 (x + 1)}$ .

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

Step 4.3

Reorder the factors of $(x + 5) (3 (x + 1))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 x (x + 5) - 2 \cdot 3 (x + 1)$ .

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Step 6.1.1

Factor $2$ out of $2 x (x + 5)$ .

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

Step 6.1.2

Factor $2$ out of $- 2 \cdot 3 (x + 1)$ .

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

Step 6.1.3

Factor $2$ out of $2 (x (x + 5)) + 2 (- 1 \cdot 3 (x + 1))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $x$ by $x$ .

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

Step 6.4

Move $5$ to the left of $x$ .

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

Step 6.5

Multiply $- 1$ by $3$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Multiply $- 3$ by $1$ .

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

Step 6.8

Subtract $3 x$ from $5 x$ .

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

Step 6.9

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 6.9.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 6.9.2

Write the factored form using these integers.

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

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