Trigonometry Examples

$| \frac{4}{3} - \frac{x}{3} | = \frac{5}{3}$

Step 1

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

$\frac{4}{3} - \frac{x}{3} = \pm \frac{5}{3}$

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.

$\frac{4}{3} - \frac{x}{3} = \frac{5}{3}$

Step 2.2

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

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Step 2.2.1

Subtract $\frac{4}{3}$ from both sides of the equation.

$- \frac{x}{3} = \frac{5}{3} - \frac{4}{3}$

Step 2.2.2

Combine the numerators over the common denominator.

$- \frac{x}{3} = \frac{5 - 4}{3}$

Step 2.2.3

Subtract $4$ from $5$ .

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

Step 2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = 1$

Step 2.4

Divide each term in $- x = 1$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- x = 1$ by $- 1$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

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

Step 2.4.2.2

Divide $x$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 2.5

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

$\frac{4}{3} - \frac{x}{3} = - \frac{5}{3}$

Step 2.6

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

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Step 2.6.1

Subtract $\frac{4}{3}$ from both sides of the equation.

$- \frac{x}{3} = - \frac{5}{3} - \frac{4}{3}$

Step 2.6.2

Combine the numerators over the common denominator.

$- \frac{x}{3} = \frac{- 5 - 4}{3}$

Step 2.6.3

Subtract $4$ from $- 5$ .

$- \frac{x}{3} = \frac{- 9}{3}$

Step 2.6.4

Divide $- 9$ by $3$ .

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

Step 2.7

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{x}{3}) = - 3 \cdot - 3$

Step 2.8

Simplify both sides of the equation.

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Step 2.8.1

Simplify the left side.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

Cancel the common factor of $3$ .

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Step 2.8.1.1.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

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

Step 2.8.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- x}{3} = - 3 \cdot - 3$

Step 2.8.1.1.1.3

Cancel the common factor.

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

Step 2.8.1.1.1.4

Rewrite the expression.

$- - x = - 3 \cdot - 3$

Step 2.8.1.1.2

Multiply.

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Step 2.8.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 3 \cdot - 3$

Step 2.8.1.1.2.2

Multiply $x$ by $1$ .

$x = - 3 \cdot - 3$

Step 2.8.2

Simplify the right side.

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Step 2.8.2.1

Multiply $- 3$ by $- 3$ .

$x = 9$

Step 2.9

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

$x = - 1, 9$