Pre-Algebra Examples

$\frac{\frac{6}{x + 3 + x - 4}}{x + 5 + \frac{18}{x - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{x + 3 + x - 4} \cdot \frac{1}{x + 5 + \frac{18}{x - 4}}$

Step 2

Simplify the denominator.

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Step 2.1

Add $x$ and $x$ .

$\frac{6}{2 x + 3 - 4} \cdot \frac{1}{x + 5 + \frac{18}{x - 4}}$

Step 2.2

Subtract $4$ from $3$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{x + 5 + \frac{18}{x - 4}}$

Step 3

Simplify the denominator.

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Step 3.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x (x - 4)}{x - 4} + \frac{18}{x - 4} + 5}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x (x - 4) + 18}{x - 4} + 5}$

Step 3.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x (x - 4) + 18}{x - 4} + \frac{5 (x - 4)}{x - 4}}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x (x - 4) + 18 + 5 (x - 4)}{x - 4}}$

Step 3.5

Reorder terms.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x (x - 4) + 5 (x - 4) + 18}{x - 4}}$

Step 3.6

Rewrite $\frac{x (x - 4) + 5 (x - 4) + 18}{x - 4}$ in a factored form.

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Step 3.6.1

Apply the distributive property.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x \cdot x + x \cdot - 4 + 5 (x - 4) + 18}{x - 4}}$

Step 3.6.2

Multiply $x$ by $x$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} + x \cdot - 4 + 5 (x - 4) + 18}{x - 4}}$

Step 3.6.3

Move $- 4$ to the left of $x$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} - 4 \cdot x + 5 (x - 4) + 18}{x - 4}}$

Step 3.6.4

Apply the distributive property.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} - 4 x + 5 x + 5 \cdot - 4 + 18}{x - 4}}$

Step 3.6.5

Multiply $5$ by $- 4$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} - 4 x + 5 x - 20 + 18}{x - 4}}$

Step 3.6.6

Add $- 4 x$ and $5 x$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} + x - 20 + 18}{x - 4}}$

Step 3.6.7

Add $- 20$ and $18$ .

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{x^{2} + x - 2}{x - 4}}$

Step 3.6.8

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

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Step 3.6.8.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.6.8.2

Write the factored form using these integers.

$\frac{6}{2 x - 1} \cdot \frac{1}{\frac{(x - 1) (x + 2)}{x - 4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{2 x - 1} (1 \frac{x - 4}{(x - 1) (x + 2)})$

Step 5

Multiply $\frac{x - 4}{(x - 1) (x + 2)}$ by $1$ .

$\frac{6}{2 x - 1} \cdot \frac{x - 4}{(x - 1) (x + 2)}$

Step 6

Multiply $\frac{6}{2 x - 1}$ by $\frac{x - 4}{(x - 1) (x + 2)}$ .

$\frac{6 (x - 4)}{(2 x - 1) (x - 1) (x + 2)}$

Step 7

Apply the distributive property.

$\frac{6 x + 6 \cdot - 4}{(2 x - 1) (x - 1) (x + 2)}$

Step 8

Multiply $6$ by $- 4$ .

$\frac{6 x - 24}{(2 x - 1) (x - 1) (x + 2)}$

Step 9

Split the fraction $\frac{6 x - 24}{(2 x - 1) (x - 1) (x + 2)}$ into two fractions.

$\frac{6 x}{(2 x - 1) (x - 1) (x + 2)} + \frac{- 24}{(2 x - 1) (x - 1) (x + 2)}$

Step 10

Move the negative in front of the fraction.

$\frac{6 x}{(2 x - 1) (x - 1) (x + 2)} - \frac{24}{(2 x - 1) (x - 1) (x + 2)}$