Solve for x |2x-6|-4>=6

$| 2 x - 6 | - 4 \geq 6$

Write $| 2 x - 6 | - 4 \geq 6$ as a piecewise.

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

$2 x - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$2 x \geq 6$

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 $2$ .

$x \geq 3$

In the piece where $2 x - 6$ is non-negative, remove the absolute value.

$2 x - 6 - 4 \geq 6$

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

$2 x - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$2 x < 6$

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

Divide each term in $2 x < 6$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $6$ by $2$ .

$x < 3$

In the piece where $2 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 6) - 4 \geq 6$

Write as a piecewise.

${\begin{cases} 2 x - 6 - 4 \geq 6 & x \geq 3 \\ - (2 x - 6) - 4 \geq 6 & x < 3 \end{cases}$

Subtract $4$ from $- 6$ .

${\begin{cases} 2 x - 10 \geq 6 & x \geq 3 \\ - (2 x - 6) - 4 \geq 6 & x < 3 \end{cases}$

Simplify $- (2 x - 6) - 4 \geq 6$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x - 10 \geq 6 & x \geq 3 \\ - (2 x) - - 6 - 4 \geq 6 & x < 3 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 10 \geq 6 & x \geq 3 \\ - 2 x - - 6 - 4 \geq 6 & x < 3 \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 2 x - 10 \geq 6 & x \geq 3 \\ - 2 x + 6 - 4 \geq 6 & x < 3 \end{cases}$

Subtract $4$ from $6$ .

${\begin{cases} 2 x - 10 \geq 6 & x \geq 3 \\ - 2 x + 2 \geq 6 & x < 3 \end{cases}$

Solve $2 x - 10 \geq 6$ for $x$ .

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

Add $10$ to both sides of the inequality.

$2 x \geq 6 + 10$

Add $6$ and $10$ .

$2 x \geq 16$

Divide each term in $2 x \geq 16$ by $2$ and simplify.

Divide each term in $2 x \geq 16$ by $2$ .

$\frac{2 x}{2} \geq \frac{16}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{16}{2}$

Divide $x$ by $1$ .

$x \geq \frac{16}{2}$

Simplify the right side.

Divide $16$ by $2$ .

$x \geq 8$

Solve $- 2 x + 2 \geq 6$ for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 2 x \geq 6 - 2$

Subtract $2$ from $6$ .

$- 2 x \geq 4$

Divide each term in $- 2 x \geq 4$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{4}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$x \leq - 2$

Find the union of the solutions.

$x \leq - 2$ or $x \geq 8$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2 or x \geq 8$

Interval Notation:

$(- \infty, - 2] \cup [8, \infty)$

Solve for x |2x-6|-4>=6

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