Trigonometry Examples

$9 | w + 8 | + 6 > 33$

Step 1

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

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Step 1.1

Subtract $6$ from both sides of the inequality.

$9 | w + 8 | > 33 - 6$

Step 1.2

Subtract $6$ from $33$ .

$9 | w + 8 | > 27$

Step 2

Write $9 | w + 8 | > 27$ as a piecewise.

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Step 2.1

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

$w + 8 \geq 0$

Step 2.2

Subtract $8$ from both sides of the inequality.

$w \geq - 8$

Step 2.3

In the piece where $w + 8$ is non-negative, remove the absolute value.

$9 (w + 8) > 27$

Step 2.4

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

$w + 8 < 0$

Step 2.5

Subtract $8$ from both sides of the inequality.

$w < - 8$

Step 2.6

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

$9 (- (w + 8)) > 27$

Step 2.7

Write as a piecewise.

${\begin{cases} 9 (w + 8) > 27 & w \geq - 8 \\ 9 (- (w + 8)) > 27 & w < - 8 \end{cases}$

Step 2.8

Simplify $9 (w + 8) > 27$ .

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Step 2.8.1

Apply the distributive property.

${\begin{cases} 9 w + 9 \cdot 8 > 27 & w \geq - 8 \\ 9 (- (w + 8)) > 27 & w < - 8 \end{cases}$

Step 2.8.2

Multiply $9$ by $8$ .

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ 9 (- (w + 8)) > 27 & w < - 8 \end{cases}$

Step 2.9

Simplify $9 (- (w + 8)) > 27$ .

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Step 2.9.1

Apply the distributive property.

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ 9 (- w - 1 \cdot 8) > 27 & w < - 8 \end{cases}$

Step 2.9.2

Multiply $- 1$ by $8$ .

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ 9 (- w - 8) > 27 & w < - 8 \end{cases}$

Step 2.9.3

Apply the distributive property.

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ 9 (- w) + 9 \cdot - 8 > 27 & w < - 8 \end{cases}$

Step 2.9.4

Multiply $- 1$ by $9$ .

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ - 9 w + 9 \cdot - 8 > 27 & w < - 8 \end{cases}$

Step 2.9.5

Multiply $9$ by $- 8$ .

${\begin{cases} 9 w + 72 > 27 & w \geq - 8 \\ - 9 w - 72 > 27 & w < - 8 \end{cases}$

Step 3

Solve $9 w + 72 > 27$ for $w$ .

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Step 3.1

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

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Step 3.1.1

Subtract $72$ from both sides of the inequality.

$9 w > 27 - 72$

Step 3.1.2

Subtract $72$ from $27$ .

$9 w > - 45$

Step 3.2

Divide each term in $9 w > - 45$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 w > - 45$ by $9$ .

$\frac{9 w}{9} > \frac{- 45}{9}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 w}{9} > \frac{- 45}{9}$

Step 3.2.2.1.2

Divide $w$ by $1$ .

$w > \frac{- 45}{9}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 45$ by $9$ .

$w > - 5$

Step 4

Solve $- 9 w - 72 > 27$ for $w$ .

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Step 4.1

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

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Step 4.1.1

Add $72$ to both sides of the inequality.

$- 9 w > 27 + 72$

Step 4.1.2

Add $27$ and $72$ .

$- 9 w > 99$

Step 4.2

Divide each term in $- 9 w > 99$ by $- 9$ and simplify.

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Step 4.2.1

Divide each term in $- 9 w > 99$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 w}{- 9} < \frac{99}{- 9}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 9$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 9 w}{- 9} < \frac{99}{- 9}$

Step 4.2.2.1.2

Divide $w$ by $1$ .

$w < \frac{99}{- 9}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $99$ by $- 9$ .

$w < - 11$

Step 5

Find the union of the solutions.

$w < - 11$ or $w > - 5$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$w < - 11 or w > - 5$

Interval Notation:

$(- \infty, - 11) \cup (- 5, \infty)$

Step 7