Trigonometry Examples

$\frac{2}{3 m + 1} + \frac{4}{3 m - 1} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1

Simplify $\frac{2}{3 m + 1} + \frac{4}{3 m - 1}$ .

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Step 1.1

To write $\frac{2}{3 m + 1}$ as a fraction with a common denominator, multiply by $\frac{3 m - 1}{3 m - 1}$ .

$\frac{2}{3 m + 1} \cdot \frac{3 m - 1}{3 m - 1} + \frac{4}{3 m - 1} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.2

To write $\frac{4}{3 m - 1}$ as a fraction with a common denominator, multiply by $\frac{3 m + 1}{3 m + 1}$ .

$\frac{2}{3 m + 1} \cdot \frac{3 m - 1}{3 m - 1} + \frac{4}{3 m - 1} \cdot \frac{3 m + 1}{3 m + 1} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{2}{3 m + 1}$ by $\frac{3 m - 1}{3 m - 1}$ .

$\frac{2 (3 m - 1)}{(3 m + 1) (3 m - 1)} + \frac{4}{3 m - 1} \cdot \frac{3 m + 1}{3 m + 1} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.3.2

Multiply $\frac{4}{3 m - 1}$ by $\frac{3 m + 1}{3 m + 1}$ .

$\frac{2 (3 m - 1)}{(3 m + 1) (3 m - 1)} + \frac{4 (3 m + 1)}{(3 m - 1) (3 m + 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.3.3

Reorder the factors of $(3 m - 1) (3 m + 1)$ .

$\frac{2 (3 m - 1)}{(3 m + 1) (3 m - 1)} + \frac{4 (3 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2 (3 m - 1) + 4 (3 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $2$ out of $2 (3 m - 1) + 4 (3 m + 1)$ .

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Step 1.5.1.1

Factor $2$ out of $4 (3 m + 1)$ .

$\frac{2 (3 m - 1) + 2 (2 (3 m + 1))}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.1.2

Factor $2$ out of $2 (3 m - 1) + 2 (2 (3 m + 1))$ .

$\frac{2 (3 m - 1 + 2 (3 m + 1))}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.2

Apply the distributive property.

$\frac{2 (3 m - 1 + 2 (3 m) + 2 \cdot 1)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.3

Multiply $3$ by $2$ .

$\frac{2 (3 m - 1 + 6 m + 2 \cdot 1)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.4

Multiply $2$ by $1$ .

$\frac{2 (3 m - 1 + 6 m + 2)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.5

Add $3 m$ and $6 m$ .

$\frac{2 (9 m - 1 + 2)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 1.5.6

Add $- 1$ and $2$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{6 m + 8}{9 m^{2} - 1}$

Step 2

Simplify $\frac{6 m + 8}{9 m^{2} - 1}$ .

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Step 2.1

Factor $2$ out of $6 m + 8$ .

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Step 2.1.1

Factor $2$ out of $6 m$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m) + 8}{9 m^{2} - 1}$

Step 2.1.2

Factor $2$ out of $8$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m) + 2 (4)}{9 m^{2} - 1}$

Step 2.1.3

Factor $2$ out of $2 (3 m) + 2 (4)$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m + 4)}{9 m^{2} - 1}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $9 m^{2}$ as ${(3 m)}^{2}$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m + 4)}{{(3 m)}^{2} - 1}$

Step 2.2.2

Rewrite $1$ as $1^{2}$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m + 4)}{{(3 m)}^{2} - 1^{2}}$

Step 2.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 m$ and $b = 1$ .

$\frac{2 (9 m + 1)}{(3 m + 1) (3 m - 1)} = \frac{2 (3 m + 4)}{(3 m + 1) (3 m - 1)}$

Step 3

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

$m = \frac{1}{2}$

Step 4