Trigonometry Examples

$| - 8 x - 16 | > 8$

Step 1

Write $| - 8 x - 16 | > 8$ as a piecewise.

Tap for more steps...

Step 1.1

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

$- 8 x - 16 \geq 0$

Step 1.2

Solve the inequality.

Tap for more steps...

Step 1.2.1

Add $16$ to both sides of the inequality.

$- 8 x \geq 16$

Step 1.2.2

Divide each term in $- 8 x \geq 16$ by $- 8$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $- 8 x \geq 16$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{16}{- 8}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{16}{- 8}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{16}{- 8}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $16$ by $- 8$ .

$x \leq - 2$

Step 1.3

In the piece where $- 8 x - 16$ is non-negative, remove the absolute value.

$- 8 x - 16 > 8$

Step 1.4

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

$- 8 x - 16 < 0$

Step 1.5

Solve the inequality.

Tap for more steps...

Step 1.5.1

Add $16$ to both sides of the inequality.

$- 8 x < 16$

Step 1.5.2

Divide each term in $- 8 x < 16$ by $- 8$ and simplify.

Tap for more steps...

Step 1.5.2.1

Divide each term in $- 8 x < 16$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} > \frac{16}{- 8}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} > \frac{16}{- 8}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{16}{- 8}$

Step 1.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.5.2.3.1

Divide $16$ by $- 8$ .

$x > - 2$

Step 1.6

In the piece where $- 8 x - 16$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 8 x - 16) > 8$

Step 1.7

Write as a piecewise.

${\begin{cases} - 8 x - 16 > 8 & x \leq - 2 \\ - (- 8 x - 16) > 8 & x > - 2 \end{cases}$

Step 1.8

Simplify $- (- 8 x - 16) > 8$ .

Tap for more steps...

Step 1.8.1

Apply the distributive property.

${\begin{cases} - 8 x - 16 > 8 & x \leq - 2 \\ - (- 8 x) - - 16 > 8 & x > - 2 \end{cases}$

Step 1.8.2

Multiply $- 8$ by $- 1$ .

${\begin{cases} - 8 x - 16 > 8 & x \leq - 2 \\ 8 x - - 16 > 8 & x > - 2 \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 16$ .

${\begin{cases} - 8 x - 16 > 8 & x \leq - 2 \\ 8 x + 16 > 8 & x > - 2 \end{cases}$

Step 2

Solve $- 8 x - 16 > 8$ for $x$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Add $16$ to both sides of the inequality.

$- 8 x > 8 + 16$

Step 2.1.2

Add $8$ and $16$ .

$- 8 x > 24$

Step 2.2

Divide each term in $- 8 x > 24$ by $- 8$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- 8 x > 24$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} < \frac{24}{- 8}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} < \frac{24}{- 8}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{24}{- 8}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $24$ by $- 8$ .

$x < - 3$

Step 3

Solve $8 x + 16 > 8$ for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $16$ from both sides of the inequality.

$8 x > 8 - 16$

Step 3.1.2

Subtract $16$ from $8$ .

$8 x > - 8$

Step 3.2

Divide each term in $8 x > - 8$ by $8$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $8 x > - 8$ by $8$ .

$\frac{8 x}{8} > \frac{- 8}{8}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} > \frac{- 8}{8}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 8}{8}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $- 8$ by $8$ .

$x > - 1$

Step 4

Find the union of the solutions.

$x < - 3$ or $x > - 1$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 3) \cup (- 1, \infty)$

Step 6