Precalculus Examples

$| 3 x + 9 | + 9 < 14$

Step 1

Write $| 3 x + 9 | + 9 < 14$ 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.

$3 x + 9 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $9$ from both sides of the inequality.

$3 x \geq - 9$

Step 1.2.2

Divide each term in $3 x \geq - 9$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x \geq - 9$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 9}{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 9}{3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 9}{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 9$ by $3$ .

$x \geq - 3$

Step 1.3

In the piece where $3 x + 9$ is non-negative, remove the absolute value.

$3 x + 9 + 9 < 14$

Step 1.4

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

$3 x + 9 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $9$ from both sides of the inequality.

$3 x < - 9$

Step 1.5.2

Divide each term in $3 x < - 9$ by $3$ and simplify.

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Step 1.5.2.1

Divide each term in $3 x < - 9$ by $3$ .

$\frac{3 x}{3} < \frac{- 9}{3}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 9}{3}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 9}{3}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $- 9$ by $3$ .

$x < - 3$

Step 1.6

In the piece where $3 x + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x + 9) + 9 < 14$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 x + 9 + 9 < 14 & x \geq - 3 \\ - (3 x + 9) + 9 < 14 & x < - 3 \end{cases}$

Step 1.8

Add $9$ and $9$ .

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - (3 x + 9) + 9 < 14 & x < - 3 \end{cases}$

Step 1.9

Simplify $- (3 x + 9) + 9 < 14$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - (3 x) - 1 \cdot 9 + 9 < 14 & x < - 3 \end{cases}$

Step 1.9.1.2

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - 3 x - 1 \cdot 9 + 9 < 14 & x < - 3 \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $9$ .

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - 3 x - 9 + 9 < 14 & x < - 3 \end{cases}$

Step 1.9.2

Combine the opposite terms in $- 3 x - 9 + 9$ .

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Step 1.9.2.1

Add $- 9$ and $9$ .

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - 3 x + 0 < 14 & x < - 3 \end{cases}$

Step 1.9.2.2

Add $- 3 x$ and $0$ .

${\begin{cases} 3 x + 18 < 14 & x \geq - 3 \\ - 3 x < 14 & x < - 3 \end{cases}$

Step 2

Solve $3 x + 18 < 14$ when $x \geq - 3$ .

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Step 2.1

Solve $3 x + 18 < 14$ 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 $18$ from both sides of the inequality.

$3 x < 14 - 18$

Step 2.1.1.2

Subtract $18$ from $14$ .

$3 x < - 4$

Step 2.1.2

Divide each term in $3 x < - 4$ by $3$ and simplify.

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Step 2.1.2.1

Divide each term in $3 x < - 4$ by $3$ .

$\frac{3 x}{3} < \frac{- 4}{3}$

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.

$\frac{3 x}{3} < \frac{- 4}{3}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 4}{3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$x < - \frac{4}{3}$

Step 2.2

Find the intersection of $x < - \frac{4}{3}$ and $x \geq - 3$ .

$- 3 \leq x < - \frac{4}{3}$

Step 3

Solve $- 3 x < 14$ when $x < - 3$ .

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Step 3.1

Divide each term in $- 3 x < 14$ by $- 3$ and simplify.

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Step 3.1.1

Divide each term in $- 3 x < 14$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{14}{- 3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{14}{- 3}$

Step 3.1.2.1.2

Divide $x$ by $1$ .

$x > \frac{14}{- 3}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Move the negative in front of the fraction.

$x > - \frac{14}{3}$

Step 3.2

Find the intersection of $x > - \frac{14}{3}$ and $x < - 3$ .

$- \frac{14}{3} < x < - 3$

Step 4

Find the union of the solutions.

$- \frac{14}{3} < x < - \frac{4}{3}$

Step 5

Convert the inequality to interval notation.

$(- \frac{14}{3}, - \frac{4}{3})$

Step 6