Solve for x |2x|>=6

$| 2 x | \geq 6$

Step 1

Write $| 2 x | \geq 6$ as a piecewise.

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

$2 x \geq 0$

Divide each term in $2 x \geq 0$ by $2$ and simplify.

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x \geq 0$

In the piece where $2 x$ is non-negative, remove the absolute value.

$2 x \geq 6$

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

$2 x < 0$

Divide each term in $2 x < 0$ by $2$ and simplify.

Divide each term in $2 x < 0$ by $2$ .

$\frac{2 x}{2} < \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{0}{2}$

Divide $x$ by $1$ .

$x < \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x < 0$

In the piece where $2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- 2 x \geq 6$

Write as a piecewise.

${\begin{cases} 2 x \geq 6 & x \geq 0 \\ - 2 x \geq 6 & x < 0 \end{cases}$

Step 2

Divide each term in $2 x \geq 6$ by $2$ and simplify.

Divide each term in $2 x \geq 6$ by $2$ .

$\frac{2 x}{2} \geq \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{6}{2}$

Divide $x$ by $1$ .

$x \geq \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x \geq 3$

Step 3

Divide each term in $- 2 x \geq 6$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{6}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{6}{- 2}$

Simplify the right side.

Divide $6$ by $- 2$ .

$x \leq - 3$

Step 4

Find the union of the solutions.

$x \leq - 3$ or $x \geq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or x \geq 3$

Interval Notation:

$(- \infty, - 3] \cup [3, \infty)$

Step 6

Solve for x |2x|>=6

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