Trigonometry Examples

$f (x) + 2 = \frac{1}{6} \cdot | x - 3 |$

Step 1

Rewrite the equation as $\frac{1}{6} | x - 3 | = f (x) + 2$ .

$\frac{1}{6} | x - 3 | = f (x) + 2$

Step 2

Multiply both sides of the equation by $6$ .

$6 (\frac{1}{6} | x - 3 |) = 6 (f (x) + 2)$

Step 3

Simplify both sides of the equation.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Simplify $6 (\frac{1}{6} | x - 3 |)$ .

Tap for more steps...

Step 3.1.1.1

Combine $\frac{1}{6}$ and $| x - 3 |$ .

$6 \frac{| x - 3 |}{6} = 6 (f (x) + 2)$

Step 3.1.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.1.1.2.1

Cancel the common factor.

$6 \frac{| x - 3 |}{6} = 6 (f (x) + 2)$

Step 3.1.1.2.2

Rewrite the expression.

$| x - 3 | = 6 (f (x) + 2)$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $6 (f (x) + 2)$ .

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$| x - 3 | = 6 f (x) + 6 \cdot 2$

Step 3.2.1.2

Multiply $6$ by $2$ .

$| x - 3 | = 6 f (x) + 12$

Step 4

Multiply $6 f$ by $x$ .

$| x - 3 | = 6 f x + 12$

Step 5

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 3 = \pm (6 f x + 12)$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 3 = 6 f x + 12$

Step 6.2

Subtract $6 f x$ from both sides of the equation.

$x - 3 - 6 f x = 12$

Step 6.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 6.3.1

Add $3$ to both sides of the equation.

$x - 6 f x = 12 + 3$

Step 6.3.2

Add $12$ and $3$ .

$x - 6 f x = 15$

Step 6.4

Factor $x$ out of $x - 6 f x$ .

Tap for more steps...

Step 6.4.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - 6 f x = 15$

Step 6.4.2

Factor $x$ out of $- 6 f x$ .

$x \cdot 1 + x (- 6 f) = 15$

Step 6.4.3

Factor $x$ out of $x \cdot 1 + x (- 6 f)$ .

$x (1 - 6 f) = 15$

Step 6.5

Divide each term in $x (1 - 6 f) = 15$ by $1 - 6 f$ and simplify.

Tap for more steps...

Step 6.5.1

Divide each term in $x (1 - 6 f) = 15$ by $1 - 6 f$ .

$\frac{x (1 - 6 f)}{1 - 6 f} = \frac{15}{1 - 6 f}$

Step 6.5.2

Simplify the left side.

Tap for more steps...

Step 6.5.2.1

Cancel the common factor of $1 - 6 f$ .

Tap for more steps...

Step 6.5.2.1.1

Cancel the common factor.

$\frac{x (1 - 6 f)}{1 - 6 f} = \frac{15}{1 - 6 f}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{1 - 6 f}$

Step 6.6

Next, use the negative value of the $\pm$ to find the second solution.

$x - 3 = - (6 f x + 12)$

Step 6.7

Simplify $- (6 f x + 12)$ .

Tap for more steps...

Step 6.7.1

Rewrite.

$x - 3 = 0 + 0 - (6 f x + 12)$

Step 6.7.2

Simplify by adding zeros.

$x - 3 = - (6 f x + 12)$

Step 6.7.3

Apply the distributive property.

$x - 3 = - (6 f x) - 1 \cdot 12$

Step 6.7.4

Multiply.

Tap for more steps...

Step 6.7.4.1

Multiply $6$ by $- 1$ .

$x - 3 = - 6 (f x) - 1 \cdot 12$

Step 6.7.4.2

Multiply $- 1$ by $12$ .

$x - 3 = - 6 f x - 12$

Step 6.8

Add $6 f x$ to both sides of the equation.

$x - 3 + 6 f x = - 12$

Step 6.9

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 6.9.1

Add $3$ to both sides of the equation.

$x + 6 f x = - 12 + 3$

Step 6.9.2

Add $- 12$ and $3$ .

$x + 6 f x = - 9$

Step 6.10

Factor $x$ out of $x + 6 f x$ .

Tap for more steps...

Step 6.10.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 + 6 f x = - 9$

Step 6.10.2

Factor $x$ out of $6 f x$ .

$x \cdot 1 + x (6 f) = - 9$

Step 6.10.3

Factor $x$ out of $x \cdot 1 + x (6 f)$ .

$x (1 + 6 f) = - 9$

Step 6.11

Divide each term in $x (1 + 6 f) = - 9$ by $1 + 6 f$ and simplify.

Tap for more steps...

Step 6.11.1

Divide each term in $x (1 + 6 f) = - 9$ by $1 + 6 f$ .

$\frac{x (1 + 6 f)}{1 + 6 f} = \frac{- 9}{1 + 6 f}$

Step 6.11.2

Simplify the left side.

Tap for more steps...

Step 6.11.2.1

Cancel the common factor of $1 + 6 f$ .

Tap for more steps...

Step 6.11.2.1.1

Cancel the common factor.

$\frac{x (1 + 6 f)}{1 + 6 f} = \frac{- 9}{1 + 6 f}$

Step 6.11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{1 + 6 f}$

Step 6.11.3

Simplify the right side.

Tap for more steps...

Step 6.11.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{1 + 6 f}$

Step 6.12

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{15}{1 - 6 f}$

$x = - \frac{9}{1 + 6 f}$

$x = \frac{15}{1 - 6 f}$

$x = - \frac{9}{1 + 6 f}$