Trigonometry Examples

$\frac{9 x - 1}{10} - 5 x = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1

Simplify $\frac{9 x - 1}{10} - 5 x$ .

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Step 1.1

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

$\frac{9 x - 1}{10} - 5 x \cdot \frac{10}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $- 5 x$ and $\frac{10}{10}$ .

$\frac{9 x - 1}{10} + \frac{- 5 x \cdot 10}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{9 x - 1 - 5 x \cdot 10}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Multiply $10$ by $- 5$ .

$\frac{9 x - 1 - 50 x}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.3.2

Subtract $50 x$ from $9 x$ .

$\frac{- 41 x - 1}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.4

Simplify with factoring out.

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Step 1.4.1

Factor $- 1$ out of $- 41 x$ .

$\frac{- (41 x) - 1}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.4.2

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

$\frac{- (41 x) - 1 (1)}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.4.3

Factor $- 1$ out of $- (41 x) - 1 (1)$ .

$\frac{- (41 x + 1)}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Rewrite $- (41 x + 1)$ as $- 1 (41 x + 1)$ .

$\frac{- 1 (41 x + 1)}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 1.4.4.2

Move the negative in front of the fraction.

$- \frac{41 x + 1}{10} = \frac{1}{5} + \frac{8 x + 5}{15}$

Step 2

Simplify $\frac{1}{5} + \frac{8 x + 5}{15}$ .

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Step 2.1

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{41 x + 1}{10} = \frac{1}{5} \cdot \frac{3}{3} + \frac{8 x + 5}{15}$

Step 2.2

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$- \frac{41 x + 1}{10} = \frac{3}{5 \cdot 3} + \frac{8 x + 5}{15}$

Step 2.2.2

Multiply $5$ by $3$ .

$- \frac{41 x + 1}{10} = \frac{3}{15} + \frac{8 x + 5}{15}$

Step 2.3

Combine the numerators over the common denominator.

$- \frac{41 x + 1}{10} = \frac{3 + 8 x + 5}{15}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Add $3$ and $5$ .

$- \frac{41 x + 1}{10} = \frac{8 x + 8}{15}$

Step 2.4.2

Factor $8$ out of $8 x + 8$ .

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Step 2.4.2.1

Factor $8$ out of $8 x$ .

$- \frac{41 x + 1}{10} = \frac{8 (x) + 8}{15}$

Step 2.4.2.2

Factor $8$ out of $8$ .

$- \frac{41 x + 1}{10} = \frac{8 (x) + 8 (1)}{15}$

Step 2.4.2.3

Factor $8$ out of $8 (x) + 8 (1)$ .

$- \frac{41 x + 1}{10} = \frac{8 (x + 1)}{15}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.13669064$

Step 4