Trigonometry Examples

$| 3 w + 1 | - 4 < 7$

Step 1

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

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Step 1.1

Add $4$ to both sides of the inequality.

$| 3 w + 1 | < 7 + 4$

Step 1.2

Add $7$ and $4$ .

$| 3 w + 1 | < 11$

Step 2

Write $| 3 w + 1 | < 11$ 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.

$3 w + 1 \geq 0$

Step 2.2

Solve the inequality.

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Step 2.2.1

Subtract $1$ from both sides of the inequality.

$3 w \geq - 1$

Step 2.2.2

Divide each term in $3 w \geq - 1$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 w \geq - 1$ by $3$ .

$\frac{3 w}{3} \geq \frac{- 1}{3}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 w}{3} \geq \frac{- 1}{3}$

Step 2.2.2.2.1.2

Divide $w$ by $1$ .

$w \geq \frac{- 1}{3}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

$w \geq - \frac{1}{3}$

Step 2.3

In the piece where $3 w + 1$ is non-negative, remove the absolute value.

$3 w + 1 < 11$

Step 2.4

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

$3 w + 1 < 0$

Step 2.5

Solve the inequality.

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Step 2.5.1

Subtract $1$ from both sides of the inequality.

$3 w < - 1$

Step 2.5.2

Divide each term in $3 w < - 1$ by $3$ and simplify.

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Step 2.5.2.1

Divide each term in $3 w < - 1$ by $3$ .

$\frac{3 w}{3} < \frac{- 1}{3}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{3 w}{3} < \frac{- 1}{3}$

Step 2.5.2.2.1.2

Divide $w$ by $1$ .

$w < \frac{- 1}{3}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Move the negative in front of the fraction.

$w < - \frac{1}{3}$

Step 2.6

In the piece where $3 w + 1$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 w + 1) < 11$

Step 2.7

Write as a piecewise.

${\begin{cases} 3 w + 1 < 11 & w \geq - \frac{1}{3} \\ - (3 w + 1) < 11 & w < - \frac{1}{3} \end{cases}$

Step 2.8

Simplify $- (3 w + 1) < 11$ .

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Step 2.8.1

Apply the distributive property.

${\begin{cases} 3 w + 1 < 11 & w \geq - \frac{1}{3} \\ - (3 w) - 1 \cdot 1 < 11 & w < - \frac{1}{3} \end{cases}$

Step 2.8.2

Multiply $3$ by $- 1$ .

${\begin{cases} 3 w + 1 < 11 & w \geq - \frac{1}{3} \\ - 3 w - 1 \cdot 1 < 11 & w < - \frac{1}{3} \end{cases}$

Step 2.8.3

Multiply $- 1$ by $1$ .

${\begin{cases} 3 w + 1 < 11 & w \geq - \frac{1}{3} \\ - 3 w - 1 < 11 & w < - \frac{1}{3} \end{cases}$

Step 3

Solve $3 w + 1 < 11$ when $w \geq - \frac{1}{3}$ .

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Step 3.1

Solve $3 w + 1 < 11$ for $w$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $1$ from both sides of the inequality.

$3 w < 11 - 1$

Step 3.1.1.2

Subtract $1$ from $11$ .

$3 w < 10$

Step 3.1.2

Divide each term in $3 w < 10$ by $3$ and simplify.

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Step 3.1.2.1

Divide each term in $3 w < 10$ by $3$ .

$\frac{3 w}{3} < \frac{10}{3}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{3 w}{3} < \frac{10}{3}$

Step 3.1.2.2.1.2

Divide $w$ by $1$ .

$w < \frac{10}{3}$

Step 3.2

Find the intersection of $w < \frac{10}{3}$ and $w \geq - \frac{1}{3}$ .

$- \frac{1}{3} \leq w < \frac{10}{3}$

Step 4

Solve $- 3 w - 1 < 11$ when $w < - \frac{1}{3}$ .

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Step 4.1

Solve $- 3 w - 1 < 11$ for $w$ .

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Step 4.1.1

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

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Step 4.1.1.1

Add $1$ to both sides of the inequality.

$- 3 w < 11 + 1$

Step 4.1.1.2

Add $11$ and $1$ .

$- 3 w < 12$

Step 4.1.2

Divide each term in $- 3 w < 12$ by $- 3$ and simplify.

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Step 4.1.2.1

Divide each term in $- 3 w < 12$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 w}{- 3} > \frac{12}{- 3}$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 w}{- 3} > \frac{12}{- 3}$

Step 4.1.2.2.1.2

Divide $w$ by $1$ .

$w > \frac{12}{- 3}$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $12$ by $- 3$ .

$w > - 4$

Step 4.2

Find the intersection of $w > - 4$ and $w < - \frac{1}{3}$ .

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

Step 5

Find the union of the solutions.

$- 4 < w < \frac{10}{3}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$- 4 < w < \frac{10}{3}$

Interval Notation:

$(- 4, \frac{10}{3})$

Step 7