Trigonometry Examples

$| \frac{3 x - 1}{5} | = 1$

Step 1

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

$\frac{3 x - 1}{5} = \pm 1$

Step 2

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

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Step 2.1

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

$\frac{3 x - 1}{5} = 1$

Step 2.2

Multiply both sides by $5$ .

$\frac{3 x - 1}{5} \cdot 5 = 1 \cdot 5$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $5$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{3 x - 1}{5} \cdot 5 = 1 \cdot 5$

Step 2.3.1.1.2

Rewrite the expression.

$3 x - 1 = 1 \cdot 5$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $5$ by $1$ .

$3 x - 1 = 5$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Add $1$ to both sides of the equation.

$3 x = 5 + 1$

Step 2.4.1.2

Add $5$ and $1$ .

$3 x = 6$

Step 2.4.2

Divide each term in $3 x = 6$ by $3$ and simplify.

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Step 2.4.2.1

Divide each term in $3 x = 6$ by $3$ .

$\frac{3 x}{3} = \frac{6}{3}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{6}{3}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{3}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $6$ by $3$ .

$x = 2$

Step 2.5

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

$\frac{3 x - 1}{5} = - 1$

Step 2.6

Multiply both sides by $5$ .

$\frac{3 x - 1}{5} \cdot 5 = - 1 \cdot 5$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

Cancel the common factor of $5$ .

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Step 2.7.1.1.1

Cancel the common factor.

$\frac{3 x - 1}{5} \cdot 5 = - 1 \cdot 5$

Step 2.7.1.1.2

Rewrite the expression.

$3 x - 1 = - 1 \cdot 5$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Multiply $- 1$ by $5$ .

$3 x - 1 = - 5$

Step 2.8

Solve for $x$ .

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Step 2.8.1

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

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Step 2.8.1.1

Add $1$ to both sides of the equation.

$3 x = - 5 + 1$

Step 2.8.1.2

Add $- 5$ and $1$ .

$3 x = - 4$

Step 2.8.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 2.8.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $3$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 2.9

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

$x = 2, - \frac{4}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2, - \frac{4}{3}$

Decimal Form:

$x = 2, - 1.\overline{3}$

Mixed Number Form:

$x = 2, - 1 \frac{1}{3}$