Trigonometry Examples

$3 - | 4 - n | > 1$

Step 1

Write $3 - | 4 - n | > 1$ 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.

$4 - n \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $4$ from both sides of the inequality.

$- n \geq - 4$

Step 1.2.2

Divide each term in $- n \geq - 4$ by $- 1$ and simplify.

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Step 1.2.2.1

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

$\frac{- n}{- 1} \leq \frac{- 4}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} \leq \frac{- 4}{- 1}$

Step 1.2.2.2.2

Divide $n$ by $1$ .

$n \leq \frac{- 4}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 4$ by $- 1$ .

$n \leq 4$

Step 1.3

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

$3 - (4 - n) > 1$

Step 1.4

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

$4 - n < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $4$ from both sides of the inequality.

$- n < - 4$

Step 1.5.2

Divide each term in $- n < - 4$ by $- 1$ and simplify.

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Step 1.5.2.1

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

$\frac{- n}{- 1} > \frac{- 4}{- 1}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} > \frac{- 4}{- 1}$

Step 1.5.2.2.2

Divide $n$ by $1$ .

$n > \frac{- 4}{- 1}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $- 4$ by $- 1$ .

$n > 4$

Step 1.6

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

$3 - - (4 - n) > 1$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 - (4 - n) > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.8

Simplify $3 - (4 - n) > 1$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 3 - 1 \cdot 4 - - n > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.8.1.2

Multiply $- 1$ by $4$ .

${\begin{cases} 3 - 4 - - n > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.8.1.3

Multiply $- - n$ .

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Step 1.8.1.3.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} 3 - 4 + 1 n > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.8.1.3.2

Multiply $n$ by $1$ .

${\begin{cases} 3 - 4 + n > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.8.2

Subtract $4$ from $3$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - - (4 - n) > 1 & n > 4 \end{cases}$

Step 1.9

Simplify $3 - - (4 - n) > 1$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - (- 1 \cdot 4 - - n) > 1 & n > 4 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $4$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - (- 4 - - n) > 1 & n > 4 \end{cases}$

Step 1.9.1.3

Multiply $- - n$ .

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Step 1.9.1.3.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - (- 4 + 1 n) > 1 & n > 4 \end{cases}$

Step 1.9.1.3.2

Multiply $n$ by $1$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - (- 4 + n) > 1 & n > 4 \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 - - 4 - n > 1 & n > 4 \end{cases}$

Step 1.9.1.5

Multiply $- 1$ by $- 4$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 3 + 4 - n > 1 & n > 4 \end{cases}$

Step 1.9.2

Add $3$ and $4$ .

${\begin{cases} - 1 + n > 1 & n \leq 4 \\ 7 - n > 1 & n > 4 \end{cases}$

Step 2

Solve $- 1 + n > 1$ when $n \leq 4$ .

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Step 2.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 2.1.1

Add $1$ to both sides of the inequality.

$n > 1 + 1$

Step 2.1.2

Add $1$ and $1$ .

$n > 2$

Step 2.2

Find the intersection of $n > 2$ and $n \leq 4$ .

$2 < n \leq 4$

Step 3

Solve $7 - n > 1$ when $n > 4$ .

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Step 3.1

Solve $7 - n > 1$ for $n$ .

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Step 3.1.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 3.1.1.1

Subtract $7$ from both sides of the inequality.

$- n > 1 - 7$

Step 3.1.1.2

Subtract $7$ from $1$ .

$- n > - 6$

Step 3.1.2

Divide each term in $- n > - 6$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- n > - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} < \frac{- 6}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} < \frac{- 6}{- 1}$

Step 3.1.2.2.2

Divide $n$ by $1$ .

$n < \frac{- 6}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 6$ by $- 1$ .

$n < 6$

Step 3.2

Find the intersection of $n < 6$ and $n > 4$ .

$4 < n < 6$

Step 4

Find the union of the solutions.

$2 < n < 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$2 < n < 6$

Interval Notation:

$(2, 6)$

Step 6