Trigonometry Examples

$| \frac{n}{4} | > 3$

Step 1

Write $| \frac{n}{4} | > 3$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{n}{4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $4$ .

$\frac{n}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{n}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.2

Rewrite the expression.

$n \geq 0 \cdot 4$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $4$ .

$n \geq 0$

Step 1.3

In the piece where $\frac{n}{4}$ is non-negative, remove the absolute value.

$\frac{n}{4} > 3$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{n}{4} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $4$ .

$\frac{n}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Cancel the common factor of $4$ .

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Step 1.5.2.1.1.1

Cancel the common factor.

$\frac{n}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.2

Rewrite the expression.

$n < 0 \cdot 4$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $4$ .

$n < 0$

Step 1.6

In the piece where $\frac{n}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{n}{4} > 3$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{n}{4} > 3 & n \geq 0 \\ - \frac{n}{4} > 3 & n < 0 \end{cases}$

Step 2

Solve $\frac{n}{4} > 3$ for $n$ .

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Step 2.1

Multiply both sides by $4$ .

$\frac{n}{4} \cdot 4 > 3 \cdot 4$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{n}{4} \cdot 4 > 3 \cdot 4$

Step 2.2.1.1.2

Rewrite the expression.

$n > 3 \cdot 4$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $3$ by $4$ .

$n > 12$

Step 3

Solve $- \frac{n}{4} > 3$ for $n$ .

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Step 3.1

Divide each term in $- \frac{n}{4} > 3$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \frac{n}{4} > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{n}{4}}{- 1} < \frac{3}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{n}{4}}{1} < \frac{3}{- 1}$

Step 3.1.2.2

Divide $\frac{n}{4}$ by $1$ .

$\frac{n}{4} < \frac{3}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $3$ by $- 1$ .

$\frac{n}{4} < - 3$

Step 3.2

Multiply both sides by $4$ .

$\frac{n}{4} \cdot 4 < - 3 \cdot 4$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{n}{4} \cdot 4 < - 3 \cdot 4$

Step 3.3.1.1.2

Rewrite the expression.

$n < - 3 \cdot 4$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 3$ by $4$ .

$n < - 12$

Step 4

Find the union of the solutions.

$n < - 12$ or $n > 12$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 12) \cup (12, \infty)$

Step 6