Trigonometry Examples

$3 | x + 5 | + 4 < 16$

Step 1

Write $3 | x + 5 | + 4 < 16$ 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.

$x + 5 \geq 0$

Step 1.2

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

Step 1.3

In the piece where $x + 5$ is non-negative, remove the absolute value.

$3 (x + 5) + 4 < 16$

Step 1.4

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

$x + 5 < 0$

Step 1.5

Subtract $5$ from both sides of the inequality.

$x < - 5$

Step 1.6

In the piece where $x + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$3 (- (x + 5)) + 4 < 16$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 (x + 5) + 4 < 16 & x \geq - 5 \\ 3 (- (x + 5)) + 4 < 16 & x < - 5 \end{cases}$

Step 1.8

Simplify $3 (x + 5) + 4 < 16$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 3 x + 3 \cdot 5 + 4 < 16 & x \geq - 5 \\ 3 (- (x + 5)) + 4 < 16 & x < - 5 \end{cases}$

Step 1.8.1.2

Multiply $3$ by $5$ .

${\begin{cases} 3 x + 15 + 4 < 16 & x \geq - 5 \\ 3 (- (x + 5)) + 4 < 16 & x < - 5 \end{cases}$

Step 1.8.2

Add $15$ and $4$ .

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ 3 (- (x + 5)) + 4 < 16 & x < - 5 \end{cases}$

Step 1.9

Simplify $3 (- (x + 5)) + 4 < 16$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ 3 (- x - 1 \cdot 5) + 4 < 16 & x < - 5 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $5$ .

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ 3 (- x - 5) + 4 < 16 & x < - 5 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ 3 (- x) + 3 \cdot - 5 + 4 < 16 & x < - 5 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $3$ .

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ - 3 x + 3 \cdot - 5 + 4 < 16 & x < - 5 \end{cases}$

Step 1.9.1.5

Multiply $3$ by $- 5$ .

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ - 3 x - 15 + 4 < 16 & x < - 5 \end{cases}$

Step 1.9.2

Add $- 15$ and $4$ .

${\begin{cases} 3 x + 19 < 16 & x \geq - 5 \\ - 3 x - 11 < 16 & x < - 5 \end{cases}$

Step 2

Solve $3 x + 19 < 16$ when $x \geq - 5$ .

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Step 2.1

Solve $3 x + 19 < 16$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $19$ from both sides of the inequality.

$3 x < 16 - 19$

Step 2.1.1.2

Subtract $19$ from $16$ .

$3 x < - 3$

Step 2.1.2

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

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Step 2.1.2.1

Divide each term in $3 x < - 3$ by $3$ .

$\frac{3 x}{3} < \frac{- 3}{3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 3}{3}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 3}{3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $- 3$ by $3$ .

$x < - 1$

Step 2.2

Find the intersection of $x < - 1$ and $x \geq - 5$ .

$- 5 \leq x < - 1$

Step 3

Solve $- 3 x - 11 < 16$ when $x < - 5$ .

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Step 3.1

Solve $- 3 x - 11 < 16$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $11$ to both sides of the inequality.

$- 3 x < 16 + 11$

Step 3.1.1.2

Add $16$ and $11$ .

$- 3 x < 27$

Step 3.1.2

Divide each term in $- 3 x < 27$ by $- 3$ and simplify.

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Step 3.1.2.1

Divide each term in $- 3 x < 27$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{27}{- 3}$

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.

$\frac{- 3 x}{- 3} > \frac{27}{- 3}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{27}{- 3}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $27$ by $- 3$ .

$x > - 9$

Step 3.2

Find the intersection of $x > - 9$ and $x < - 5$ .

$- 9 < x < - 5$

Step 4

Find the union of the solutions.

$- 9 < x < - 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < - 1$

Interval Notation:

$(- 9, - 1)$

Step 6