Algebra Examples

$\frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $15 x$ .

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Step 1.1

Multiply $\frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$ by $\frac{15 x}{15 x}$ .

$\frac{15 x}{15 x} \cdot \frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$

Step 1.2

Combine.

$\frac{15 x (1 - \frac{x}{5})}{15 x (\frac{x - 8}{15} + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{15 x \cdot 1 + 15 x (- \frac{x}{5})}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $5$ .

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Step 3.1.1

Move the leading negative in $- \frac{x}{5}$ into the numerator.

$\frac{15 x \cdot 1 + 15 x \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.1.2

Factor $5$ out of $15 x$ .

$\frac{15 x \cdot 1 + 5 (3 x) \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.1.3

Cancel the common factor.

$\frac{15 x \cdot 1 + 5 (3 x) \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.1.4

Rewrite the expression.

$\frac{15 x \cdot 1 + 3 x (- x)}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.2

Multiply $- 1$ by $3$ .

$\frac{15 x \cdot 1 - 3 x \cdot x}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.3

Raise $x$ to the power of $1$ .

$\frac{15 x \cdot 1 - 3 (x^{1} x)}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.4

Raise $x$ to the power of $1$ .

$\frac{15 x \cdot 1 - 3 (x^{1} x^{1})}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 x \cdot 1 - 3 x^{1 + 1}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.6

Add $1$ and $1$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.7

Cancel the common factor of $15$ .

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Step 3.7.1

Factor $15$ out of $15 x$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{15 (x) \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.7.2

Cancel the common factor.

$\frac{15 x \cdot 1 - 3 x^{2}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3.7.3

Rewrite the expression.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + 15 x \frac{1}{x}}$

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Factor $x$ out of $15 x$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Step 3.8.2

Cancel the common factor.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Step 3.8.3

Rewrite the expression.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + 15}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3 x$ out of $15 x \cdot 1 - 3 x^{2}$ .

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Step 4.1.1

Factor $3 x$ out of $15 x \cdot 1$ .

$\frac{3 x (5 \cdot 1) - 3 x^{2}}{x (x - 8) + 15}$

Step 4.1.2

Factor $3 x$ out of $- 3 x^{2}$ .

$\frac{3 x (5 \cdot 1) + 3 x (- x)}{x (x - 8) + 15}$

Step 4.1.3

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

$\frac{3 x (5 \cdot 1 - x)}{x (x - 8) + 15}$

Step 4.2

Multiply $5$ by $1$ .

$\frac{3 x (5 - x)}{x (x - 8) + 15}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{3 x (5 - x)}{x \cdot x + x \cdot - 8 + 15}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{3 x (5 - x)}{x^{2} + x \cdot - 8 + 15}$

Step 5.3

Move $- 8$ to the left of $x$ .

$\frac{3 x (5 - x)}{x^{2} - 8 \cdot x + 15}$

Step 5.4

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 5.4.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 5.4.2

Write the factored form using these integers.

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $5 - x$ and $x - 5$ .

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Step 6.1.1

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 6.1.2

Factor $- 1$ out of $- x$ .

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

Step 6.1.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Multiply $- 1$ by $3$ .

$\frac{- 3 x}{x - 3}$

Step 6.2.2

Move the negative in front of the fraction.

$- \frac{3 x}{x - 3}$