Algebra Examples

$2 | 2 x - 2 | > 20$

Step 1

Solve

$x < - 4 or x > 6$ .

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Step 1.1

Write

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

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Step 1.1.1

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

$2 x - 2 \geq 0$

Step 1.1.2

Solve the inequality.

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Step 1.1.2.1

Add

$2$ to both sides of the inequality.

$2 x \geq 2$

Step 1.1.2.2

Divide each term in

$2 x \geq 2$ by

$2$ and simplify.

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Step 1.1.2.2.1

Divide each term in

$2 x \geq 2$ by

$2$ .

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

Step 1.1.2.2.2

Simplify the left side.

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Step 1.1.2.2.2.1

Cancel the common factor of

$2$ .

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Step 1.1.2.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 1.1.2.2.3

Simplify the right side.

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Step 1.1.2.2.3.1

Divide

$2$ by

$2$ .

$x \geq 1$

Step 1.1.3

In the piece where

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

$2 (2 x - 2) > 20$

Step 1.1.4

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

$2 x - 2 < 0$

Step 1.1.5

Solve the inequality.

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Step 1.1.5.1

Add

$2$ to both sides of the inequality.

$2 x < 2$

Step 1.1.5.2

Divide each term in

$2 x < 2$ by

$2$ and simplify.

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Step 1.1.5.2.1

Divide each term in

$2 x < 2$ by

$2$ .

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

Step 1.1.5.2.2

Simplify the left side.

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Step 1.1.5.2.2.1

Cancel the common factor of

$2$ .

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Step 1.1.5.2.2.1.1

Cancel the common factor.

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

Step 1.1.5.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 1.1.5.2.3

Simplify the right side.

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Step 1.1.5.2.3.1

Divide

$2$ by

$2$ .

$x < 1$

Step 1.1.6

In the piece where

$2 x - 2$ is negative, remove the absolute value and multiply by

$- 1$ .

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

Step 1.1.7

Write as a piecewise.

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

Step 1.1.8

Simplify

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

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Step 1.1.8.1

Apply the distributive property.

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

Step 1.1.8.2

Multiply

$2$ by

$2$ .

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

Step 1.1.8.3

Multiply

$2$ by

$- 2$ .

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

Step 1.1.9

Simplify

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

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Step 1.1.9.1

Apply the distributive property.

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

Step 1.1.9.2

Multiply

$2$ by

$- 1$ .

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

Step 1.1.9.3

Multiply

$- 1$ by

$- 2$ .

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

Step 1.1.9.4

Apply the distributive property.

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

Step 1.1.9.5

Multiply

$- 2$ by

$2$ .

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

Step 1.1.9.6

Multiply

$2$ by

$2$ .

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

Step 1.2

Solve

$4 x - 4 > 20$ for

$x$ .

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Step 1.2.1

Move all terms not containing

$x$ to the right side of the inequality.

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Step 1.2.1.1

Add

$4$ to both sides of the inequality.

$4 x > 20 + 4$

Step 1.2.1.2

Add

$20$ and

$4$ .

$4 x > 24$

Step 1.2.2

Divide each term in

$4 x > 24$ by

$4$ and simplify.

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Step 1.2.2.1

Divide each term in

$4 x > 24$ by

$4$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of

$4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide

$24$ by

$4$ .

$x > 6$

Step 1.3

Solve

$- 4 x + 4 > 20$ for

$x$ .

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Step 1.3.1

Move all terms not containing

$x$ to the right side of the inequality.

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Step 1.3.1.1

Subtract

$4$ from both sides of the inequality.

$- 4 x > 20 - 4$

Step 1.3.1.2

Subtract

$4$ from

$20$ .

$- 4 x > 16$

Step 1.3.2

Divide each term in

$- 4 x > 16$ by

$- 4$ and simplify.

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Step 1.3.2.1

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.

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

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Cancel the common factor of

$- 4$ .

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Step 1.3.2.2.1.1

Cancel the common factor.

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

Step 1.3.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 1.3.2.3

Simplify the right side.

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Step 1.3.2.3.1

Divide

$16$ by

$- 4$ .

$x < - 4$

Step 1.4

Find the union of the solutions.

$x < - 4$ or

$x > 6$

$x < - 4$ 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