Convert to Interval Notation |3x-6|>12

$| 3 x - 6 | > 12$

Step 1

Write $| 3 x - 6 | > 12$ as a piecewise.

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

$3 x - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$3 x \geq 6$

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

Divide each term in $3 x \geq 6$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $6$ by $3$ .

$x \geq 2$

In the piece where $3 x - 6$ is non-negative, remove the absolute value.

$3 x - 6 > 12$

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

$3 x - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$3 x < 6$

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 $3$ .

$x < 2$

In the piece where $3 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x - 6) > 12$

Write as a piecewise.

${\begin{cases} 3 x - 6 > 12 & x \geq 2 \\ - (3 x - 6) > 12 & x < 2 \end{cases}$

Simplify $- (3 x - 6) > 12$ .

Apply the distributive property.

${\begin{cases} 3 x - 6 > 12 & x \geq 2 \\ - (3 x) - - 6 > 12 & x < 2 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x - 6 > 12 & x \geq 2 \\ - 3 x - - 6 > 12 & x < 2 \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 3 x - 6 > 12 & x \geq 2 \\ - 3 x + 6 > 12 & x < 2 \end{cases}$

Step 2

Solve $3 x - 6 > 12$ for $x$ .

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

Add $6$ to both sides of the inequality.

$3 x > 12 + 6$

Add $12$ and $6$ .

$3 x > 18$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $18$ by $3$ .

$x > 6$

Step 3

Solve $- 3 x + 6 > 12$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 3 x > 12 - 6$

Subtract $6$ from $12$ .

$- 3 x > 6$

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 $- 3$ .

$x < - 2$

Step 4

Find the union of the solutions.

$x < - 2$ or $x > 6$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 2) \cup (6, \infty)$

Step 6

Convert to Interval Notation |3x-6|>12

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