Pre-Algebra Examples

$\frac{1}{7 x^{2} - 7 x} + \frac{6}{7 x^{2} + 14 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $7 x$ out of $7 x^{2} - 7 x$ .

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Step 1.1.1

Factor $7 x$ out of $7 x^{2}$ .

$\frac{1}{7 x (x) - 7 x} + \frac{6}{7 x^{2} + 14 x}$

Step 1.1.2

Factor $7 x$ out of $- 7 x$ .

$\frac{1}{7 x (x) + 7 x (- 1)} + \frac{6}{7 x^{2} + 14 x}$

Step 1.1.3

Factor $7 x$ out of $7 x (x) + 7 x (- 1)$ .

$\frac{1}{7 x (x - 1)} + \frac{6}{7 x^{2} + 14 x}$

Step 1.2

Factor $7 x$ out of $7 x^{2} + 14 x$ .

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Step 1.2.1

Factor $7 x$ out of $7 x^{2}$ .

$\frac{1}{7 x (x - 1)} + \frac{6}{7 x (x) + 14 x}$

Step 1.2.2

Factor $7 x$ out of $14 x$ .

$\frac{1}{7 x (x - 1)} + \frac{6}{7 x (x) + 7 x (2)}$

Step 1.2.3

Factor $7 x$ out of $7 x (x) + 7 x (2)$ .

$\frac{1}{7 x (x - 1)} + \frac{6}{7 x (x + 2)}$

Step 2

To write $\frac{1}{7 x (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{1}{7 x (x - 1)} \cdot \frac{x + 2}{x + 2} + \frac{6}{7 x (x + 2)}$

Step 3

To write $\frac{6}{7 x (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{1}{7 x (x - 1)} \cdot \frac{x + 2}{x + 2} + \frac{6}{7 x (x + 2)} \cdot \frac{x - 1}{x - 1}$

Step 4

Write each expression with a common denominator of $7 x (x - 1) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{7 x (x - 1)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{x + 2}{7 x (x - 1) (x + 2)} + \frac{6}{7 x (x + 2)} \cdot \frac{x - 1}{x - 1}$

Step 4.2

Multiply $\frac{6}{7 x (x + 2)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x + 2}{7 x (x - 1) (x + 2)} + \frac{6 (x - 1)}{7 x (x + 2) (x - 1)}$

Step 4.3

Reorder the factors of $7 x (x - 1) (x + 2)$ .

$\frac{x + 2}{7 x (x + 2) (x - 1)} + \frac{6 (x - 1)}{7 x (x + 2) (x - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x + 2 + 6 (x - 1)}{7 x (x + 2) (x - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x + 2 + 6 x + 6 \cdot - 1}{7 x (x + 2) (x - 1)}$

Step 6.2

Multiply $6$ by $- 1$ .

$\frac{x + 2 + 6 x - 6}{7 x (x + 2) (x - 1)}$

Step 6.3

Add $x$ and $6 x$ .

$\frac{7 x + 2 - 6}{7 x (x + 2) (x - 1)}$

Step 6.4

Subtract $6$ from $2$ .

$\frac{7 x - 4}{7 x (x + 2) (x - 1)}$