Trigonometry Examples

$| \frac{x}{2} - 1 | > 5$

Step 1

Write $| \frac{x}{2} - 1 | > 5$ 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.

$\frac{x}{2} - 1 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $1$ to both sides of the inequality.

$\frac{x}{2} \geq 1$

Step 1.2.2

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq 1 \cdot 2$

Step 1.2.3

Simplify.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq 1 \cdot 2$

Step 1.2.3.1.1.2

Rewrite the expression.

$x \geq 1 \cdot 2$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $1$ .

$x \geq 2$

Step 1.3

In the piece where $\frac{x}{2} - 1$ is non-negative, remove the absolute value.

$\frac{x}{2} - 1 > 5$

Step 1.4

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

$\frac{x}{2} - 1 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $1$ to both sides of the inequality.

$\frac{x}{2} < 1$

Step 1.5.2

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 < 1 \cdot 2$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Simplify the left side.

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Step 1.5.3.1.1

Cancel the common factor of $2$ .

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Step 1.5.3.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 < 1 \cdot 2$

Step 1.5.3.1.1.2

Rewrite the expression.

$x < 1 \cdot 2$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Multiply $2$ by $1$ .

$x < 2$

Step 1.6

In the piece where $\frac{x}{2} - 1$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{x}{2} - 1) > 5$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{x}{2} - 1 > 5 & x \geq 2 \\ - (\frac{x}{2} - 1) > 5 & x < 2 \end{cases}$

Step 1.8

Simplify $- (\frac{x}{2} - 1) > 5$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} \frac{x}{2} - 1 > 5 & x \geq 2 \\ - \frac{x}{2} - - 1 > 5 & x < 2 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $- 1$ .

${\begin{cases} \frac{x}{2} - 1 > 5 & x \geq 2 \\ - \frac{x}{2} + 1 > 5 & x < 2 \end{cases}$

Step 2

Solve $\frac{x}{2} - 1 > 5$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $1$ to both sides of the inequality.

$\frac{x}{2} > 5 + 1$

Step 2.1.2

Add $5$ and $1$ .

$\frac{x}{2} > 6$

Step 2.2

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 > 6 \cdot 2$

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 $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 > 6 \cdot 2$

Step 2.3.1.1.2

Rewrite the expression.

$x > 6 \cdot 2$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $6$ by $2$ .

$x > 12$

Step 3

Solve $- \frac{x}{2} + 1 > 5$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1$ from both sides of the inequality.

$- \frac{x}{2} > 5 - 1$

Step 3.1.2

Subtract $1$ from $5$ .

$- \frac{x}{2} > 4$

Step 3.2

Divide each term in $- \frac{x}{2} > 4$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \frac{x}{2} > 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{2}}{- 1} < \frac{4}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} < \frac{4}{- 1}$

Step 3.2.2.2

Divide $\frac{x}{2}$ by $1$ .

$\frac{x}{2} < \frac{4}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $4$ by $- 1$ .

$\frac{x}{2} < - 4$

Step 3.3

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 < - 4 \cdot 2$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $2$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 < - 4 \cdot 2$

Step 3.4.1.1.2

Rewrite the expression.

$x < - 4 \cdot 2$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 4$ by $2$ .

$x < - 8$

Step 4

Find the union of the solutions.

$x < - 8$ or $x > 12$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 8 or x > 12$

Interval Notation:

$(- \infty, - 8) \cup (12, \infty)$

Step 6