Pre-Algebra Examples

$\frac{5}{3 x - 21} + \frac{x}{x^{2} - 49}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x - 21$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{5}{3 (x) - 21} + \frac{x}{x^{2} - 49}$

Step 1.1.2

Factor $3$ out of $- 21$ .

$\frac{5}{3 x + 3 \cdot - 7} + \frac{x}{x^{2} - 49}$

Step 1.1.3

Factor $3$ out of $3 x + 3 \cdot - 7$ .

$\frac{5}{3 (x - 7)} + \frac{x}{x^{2} - 49}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $49$ as $7^{2}$ .

$\frac{5}{3 (x - 7)} + \frac{x}{x^{2} - 7^{2}}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{5}{3 (x - 7)} + \frac{x}{(x + 7) (x - 7)}$

Step 2

To write $\frac{5}{3 (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{5}{3 (x - 7)} \cdot \frac{x + 7}{x + 7} + \frac{x}{(x + 7) (x - 7)}$

Step 3

To write $\frac{x}{(x + 7) (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{3 (x - 7)} \cdot \frac{x + 7}{x + 7} + \frac{x}{(x + 7) (x - 7)} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $3 (x - 7) (x + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5}{3 (x - 7)}$ by $\frac{x + 7}{x + 7}$ .

$\frac{5 (x + 7)}{3 (x - 7) (x + 7)} + \frac{x}{(x + 7) (x - 7)} \cdot \frac{3}{3}$

Step 4.2

Multiply $\frac{x}{(x + 7) (x - 7)}$ by $\frac{3}{3}$ .

$\frac{5 (x + 7)}{3 (x - 7) (x + 7)} + \frac{x \cdot 3}{(x + 7) (x - 7) \cdot 3}$

Step 4.3

Reorder the factors of $3 (x - 7) (x + 7)$ .

$\frac{5 (x + 7)}{3 (x + 7) (x - 7)} + \frac{x \cdot 3}{(x + 7) (x - 7) \cdot 3}$

Step 4.4

Reorder the factors of $(x + 7) (x - 7) \cdot 3$ .

$\frac{5 (x + 7)}{3 (x + 7) (x - 7)} + \frac{x \cdot 3}{3 (x + 7) (x - 7)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (x + 7) + x \cdot 3}{3 (x + 7) (x - 7)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{5 x + 5 \cdot 7 + x \cdot 3}{3 (x + 7) (x - 7)}$

Step 6.2

Multiply $5$ by $7$ .

$\frac{5 x + 35 + x \cdot 3}{3 (x + 7) (x - 7)}$

Step 6.3

Move $3$ to the left of $x$ .

$\frac{5 x + 35 + 3 \cdot x}{3 (x + 7) (x - 7)}$

Step 6.4

Add $5 x$ and $3 x$ .

$\frac{8 x + 35}{3 (x + 7) (x - 7)}$