Finite Math Examples

$| 3 x | > 18$

Step 1

Write $| 3 x | > 18$ 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.

$3 x \geq 0$

Step 1.2

Divide each term in $3 x \geq 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x \geq 0$

Step 1.3

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

$3 x > 18$

Step 1.4

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

$3 x < 0$

Step 1.5

Divide each term in $3 x < 0$ by $3$ and simplify.

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Step 1.5.1

Divide each term in $3 x < 0$ by $3$ .

$\frac{3 x}{3} < \frac{0}{3}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{0}{3}$

Step 1.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{0}{3}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $3$ .

$x < 0$

Step 1.6

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

$- 3 x > 18$

Step 1.7

Write as a piecewise.

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

Step 2

Divide each term in $3 x > 18$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x > 18$ by $3$ .

$\frac{3 x}{3} > \frac{18}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{18}{3}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x > \frac{18}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $18$ by $3$ .

$x > 6$

Step 3

Divide each term in $- 3 x > 18$ by $- 3$ and simplify.

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Step 3.1

Divide each term in $- 3 x > 18$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} < \frac{18}{- 3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} < \frac{18}{- 3}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x < \frac{18}{- 3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $18$ by $- 3$ .

$x < - 6$

Step 4

Find the union of the solutions.

$x < - 6$ or $x > 6$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 6) \cup (6, \infty)$

Step 6