Trigonometry Examples

$\frac{x + 10}{3 x - 15} - \frac{3 x + 15}{6 x - 30}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x - 15$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{x + 10}{3 (x) - 15} - \frac{3 x + 15}{6 x - 30}$

Step 1.1.2

Factor $3$ out of $- 15$ .

$\frac{x + 10}{3 x + 3 \cdot - 5} - \frac{3 x + 15}{6 x - 30}$

Step 1.1.3

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

$\frac{x + 10}{3 (x - 5)} - \frac{3 x + 15}{6 x - 30}$

Step 1.2

Cancel the common factor of $3 x + 15$ and $6 x - 30$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x) + 15}{6 x - 30}$

Step 1.2.2

Factor $3$ out of $15$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 x + 3 \cdot 5}{6 x - 30}$

Step 1.2.3

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x + 5)}{6 x - 30}$

Step 1.2.4

Cancel the common factors.

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Step 1.2.4.1

Factor $3$ out of $6 x$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x + 5)}{3 (2 x) - 30}$

Step 1.2.4.2

Factor $3$ out of $- 30$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x + 5)}{3 (2 x) + 3 (- 10)}$

Step 1.2.4.3

Factor $3$ out of $3 (2 x) + 3 (- 10)$ .

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x + 5)}{3 (2 x - 10)}$

Step 1.2.4.4

Cancel the common factor.

$\frac{x + 10}{3 (x - 5)} - \frac{3 (x + 5)}{3 (2 x - 10)}$

Step 1.2.4.5

Rewrite the expression.

$\frac{x + 10}{3 (x - 5)} - \frac{x + 5}{2 x - 10}$

Step 1.3

Factor $2$ out of $2 x - 10$ .

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Step 1.3.1

Factor $2$ out of $2 x$ .

$\frac{x + 10}{3 (x - 5)} - \frac{x + 5}{2 (x) - 10}$

Step 1.3.2

Factor $2$ out of $- 10$ .

$\frac{x + 10}{3 (x - 5)} - \frac{x + 5}{2 x + 2 \cdot - 5}$

Step 1.3.3

Factor $2$ out of $2 x + 2 \cdot - 5$ .

$\frac{x + 10}{3 (x - 5)} - \frac{x + 5}{2 (x - 5)}$

Step 2

To write $\frac{x + 10}{3 (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x + 10}{3 (x - 5)} \cdot \frac{2}{2} - \frac{x + 5}{2 (x - 5)}$

Step 3

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

$\frac{x + 10}{3 (x - 5)} \cdot \frac{2}{2} - \frac{x + 5}{2 (x - 5)} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $6 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 10}{3 (x - 5)}$ by $\frac{2}{2}$ .

$\frac{(x + 10) \cdot 2}{3 (x - 5) \cdot 2} - \frac{x + 5}{2 (x - 5)} \cdot \frac{3}{3}$

Step 4.2

Multiply $2$ by $3$ .

$\frac{(x + 10) \cdot 2}{6 (x - 5)} - \frac{x + 5}{2 (x - 5)} \cdot \frac{3}{3}$

Step 4.3

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

$\frac{(x + 10) \cdot 2}{6 (x - 5)} - \frac{(x + 5) \cdot 3}{2 (x - 5) \cdot 3}$

Step 4.4

Multiply $3$ by $2$ .

$\frac{(x + 10) \cdot 2}{6 (x - 5)} - \frac{(x + 5) \cdot 3}{6 (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 10) \cdot 2 - (x + 5) \cdot 3}{6 (x - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot 2 + 10 \cdot 2 - (x + 5) \cdot 3}{6 (x - 5)}$

Step 6.2

Move $2$ to the left of $x$ .

$\frac{2 \cdot x + 10 \cdot 2 - (x + 5) \cdot 3}{6 (x - 5)}$

Step 6.3

Multiply $10$ by $2$ .

$\frac{2 \cdot x + 20 - (x + 5) \cdot 3}{6 (x - 5)}$

Step 6.4

Apply the distributive property.

$\frac{2 x + 20 + (- x - 1 \cdot 5) \cdot 3}{6 (x - 5)}$

Step 6.5

Multiply $- 1$ by $5$ .

$\frac{2 x + 20 + (- x - 5) \cdot 3}{6 (x - 5)}$

Step 6.6

Apply the distributive property.

$\frac{2 x + 20 - x \cdot 3 - 5 \cdot 3}{6 (x - 5)}$

Step 6.7

Multiply $3$ by $- 1$ .

$\frac{2 x + 20 - 3 x - 5 \cdot 3}{6 (x - 5)}$

Step 6.8

Multiply $- 5$ by $3$ .

$\frac{2 x + 20 - 3 x - 15}{6 (x - 5)}$

Step 6.9

Subtract $3 x$ from $2 x$ .

$\frac{- x + 20 - 15}{6 (x - 5)}$

Step 6.10

Subtract $15$ from $20$ .

$\frac{- x + 5}{6 (x - 5)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

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

$\frac{- (x) + 5}{6 (x - 5)}$

Step 7.1.2

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

$\frac{- (x) - 1 (- 5)}{6 (x - 5)}$

Step 7.1.3

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

$\frac{- (x - 5)}{6 (x - 5)}$

Step 7.1.4

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

$\frac{- 1 (x - 5)}{6 (x - 5)}$

Step 7.1.5

Cancel the common factor.

$\frac{- 1 (x - 5)}{6 (x - 5)}$

Step 7.1.6

Rewrite the expression.

$\frac{- 1}{6}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{1}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{6}$

Decimal Form:

$- 0.1\overline{6}$