Convert to Set Notation 2|2x-2|>20

$2 | 2 x - 2 | > 20$

Step 1

Solve $x < - 4 or x > 6$ .

Write $2 | 2 x - 2 | > 20$ as a piecewise.

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

$2 x - 2 \geq 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$2 x \geq 2$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $2$ by $2$ .

$x \geq 1$

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

$2 (2 x - 2) > 20$

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

$2 x - 2 < 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$2 x < 2$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $2$ by $2$ .

$x < 1$

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

$2 (- (2 x - 2)) > 20$

Write as a piecewise.

${\begin{cases} 2 (2 x - 2) > 20 & x \geq 1 \\ 2 (- (2 x - 2)) > 20 & x < 1 \end{cases}$

Simplify $2 (2 x - 2) > 20$ .

Apply the distributive property.

${\begin{cases} 2 (2 x) + 2 \cdot - 2 > 20 & x \geq 1 \\ 2 (- (2 x - 2)) > 20 & x < 1 \end{cases}$

Multiply $2$ by $2$ .

${\begin{cases} 4 x + 2 \cdot - 2 > 20 & x \geq 1 \\ 2 (- (2 x - 2)) > 20 & x < 1 \end{cases}$

Multiply $2$ by $- 2$ .

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ 2 (- (2 x - 2)) > 20 & x < 1 \end{cases}$

Simplify $2 (- (2 x - 2)) > 20$ .

Apply the distributive property.

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ 2 (- (2 x) - - 2) > 20 & x < 1 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ 2 (- 2 x - - 2) > 20 & x < 1 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ 2 (- 2 x + 2) > 20 & x < 1 \end{cases}$

Apply the distributive property.

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ 2 (- 2 x) + 2 \cdot 2 > 20 & x < 1 \end{cases}$

Multiply $- 2$ by $2$ .

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ - 4 x + 2 \cdot 2 > 20 & x < 1 \end{cases}$

Multiply $2$ by $2$ .

${\begin{cases} 4 x - 4 > 20 & x \geq 1 \\ - 4 x + 4 > 20 & x < 1 \end{cases}$

Solve $4 x - 4 > 20$ for $x$ .

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

Add $4$ to both sides of the inequality.

$4 x > 20 + 4$

Add $20$ and $4$ .

$4 x > 24$

Divide each term in $4 x > 24$ by $4$ and simplify.

Divide each term in $4 x > 24$ by $4$ .

$\frac{4 x}{4} > \frac{24}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} > \frac{24}{4}$

Divide $x$ by $1$ .

$x > \frac{24}{4}$

Simplify the right side.

Divide $24$ by $4$ .

$x > 6$

Solve $- 4 x + 4 > 20$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 4 x > 20 - 4$

Subtract $4$ from $20$ .

$- 4 x > 16$

Divide each term in $- 4 x > 16$ by $- 4$ and simplify.

Divide each term in $- 4 x > 16$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{16}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{16}{- 4}$

Divide $x$ by $1$ .

$x < \frac{16}{- 4}$

Simplify the right side.

Divide $16$ by $- 4$ .

$x < - 4$

Find the union of the solutions.

$x < - 4$ or $x > 6$

Step 2

Use the inequality $x < - 4 or x > 6$ to build the set notation.

${x | x < - 4 or x > 6}$

Step 3

Convert to Set Notation 2|2x-2|>20

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