Solve the Inequality for x |3x|>6

$| 3 x | > 6$

Write $| 3 x | > 6$ as a piecewise.

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

$3 x \geq 0$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $3$ .

$x \geq 0$

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

$3 x > 6$

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

$3 x < 0$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $3$ .

$x < 0$

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

$- 3 x > 6$

Write as a piecewise.

${\begin{cases} 3 x > 6 & x \geq 0 \\ - 3 x > 6 & x < 0 \end{cases}$

Divide each term in $3 x > 6$ by $3$ and simplify.

Divide each term in $3 x > 6$ by $3$ .

$\frac{3 x}{3} > \frac{6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{6}{3}$

Divide $x$ by $1$ .

$x > \frac{6}{3}$

Simplify the right side.

Divide $6$ by $3$ .

$x > 2$

Divide each term in $- 3 x > 6$ by $- 3$ and simplify.

Divide each term in $- 3 x > 6$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} < \frac{6}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} < \frac{6}{- 3}$

Divide $x$ by $1$ .

$x < \frac{6}{- 3}$

Simplify the right side.

Divide $6$ by $- 3$ .

$x < - 2$

Find the union of the solutions.

$x < - 2$ or $x > 2$

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 2$

Interval Notation:

$(- \infty, - 2) \cup (2, \infty)$

Solve the Inequality for x |3x|>6

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