Pre-Algebra Examples

$| \frac{2 x - 4}{4} | > 1$

Step 1

Write $| \frac{2 x - 4}{4} | > 1$ as a piecewise.

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Step 1.1

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

$\frac{2 x - 4}{4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Cancel the common factor of $2 x - 4$ and $4$ .

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Step 1.2.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{4} \geq 0$

Step 1.2.1.2

Factor $2$ out of $- 4$ .

$\frac{2 x + 2 \cdot - 2}{4} \geq 0$

Step 1.2.1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{2 (x - 2)}{4} \geq 0$

Step 1.2.1.4

Cancel the common factors.

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Step 1.2.1.4.1

Factor $2$ out of $4$ .

$\frac{2 (x - 2)}{2 (2)} \geq 0$

Step 1.2.1.4.2

Cancel the common factor.

$\frac{2 (x - 2)}{2 \cdot 2} \geq 0$

Step 1.2.1.4.3

Rewrite the expression.

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

Step 1.2.2

Multiply both sides by $2$ .

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

Step 1.2.3

Simplify.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

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

Step 1.2.3.1.1.2

Rewrite the expression.

$x - 2 \geq 0 \cdot 2$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $0$ by $2$ .

$x - 2 \geq 0$

Step 1.2.4

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 1.3

In the piece where $\frac{2 x - 4}{4}$ is non-negative, remove the absolute value.

$\frac{2 x - 4}{4} > 1$

Step 1.4

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

$\frac{2 x - 4}{4} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Cancel the common factor of $2 x - 4$ and $4$ .

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Step 1.5.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{4} < 0$

Step 1.5.1.2

Factor $2$ out of $- 4$ .

$\frac{2 x + 2 \cdot - 2}{4} < 0$

Step 1.5.1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{2 (x - 2)}{4} < 0$

Step 1.5.1.4

Cancel the common factors.

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Step 1.5.1.4.1

Factor $2$ out of $4$ .

$\frac{2 (x - 2)}{2 (2)} < 0$

Step 1.5.1.4.2

Cancel the common factor.

$\frac{2 (x - 2)}{2 \cdot 2} < 0$

Step 1.5.1.4.3

Rewrite the expression.

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

Step 1.5.2

Multiply both sides by $2$ .

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Simplify the left side.

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Step 1.5.3.1.1

Cancel the common factor of $2$ .

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Step 1.5.3.1.1.1

Cancel the common factor.

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

Step 1.5.3.1.1.2

Rewrite the expression.

$x - 2 < 0 \cdot 2$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Multiply $0$ by $2$ .

$x - 2 < 0$

Step 1.5.4

Add $2$ to both sides of the inequality.

$x < 2$

Step 1.6

In the piece where $\frac{2 x - 4}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{2 x - 4}{4} > 1$

Step 1.7

Write as a piecewise.

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

Step 1.8

Cancel the common factor of $2 x - 4$ and $4$ .

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Step 1.8.1

Factor $2$ out of $2 x$ .

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

Step 1.8.2

Factor $2$ out of $- 4$ .

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

Step 1.8.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

Step 1.8.4

Cancel the common factors.

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Step 1.8.4.1

Factor $2$ out of $4$ .

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

Step 1.8.4.2

Cancel the common factor.

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

Step 1.8.4.3

Rewrite the expression.

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

Step 1.9

Cancel the common factor of $2 x - 4$ and $4$ .

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Step 1.9.1

Factor $2$ out of $2 x$ .

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

Step 1.9.2

Factor $2$ out of $- 4$ .

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

Step 1.9.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

Step 1.9.4

Cancel the common factors.

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Step 1.9.4.1

Factor $2$ out of $4$ .

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

Step 1.9.4.2

Cancel the common factor.

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

Step 1.9.4.3

Rewrite the expression.

${\begin{cases} \frac{x - 2}{2} > 1 & x \geq 2 \\ - \frac{x - 2}{2} > 1 & x < 2 \end{cases}$

Step 2

Solve $\frac{x - 2}{2} > 1$ for $x$ .

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Step 2.1

Multiply both sides by $2$ .

$\frac{x - 2}{2} \cdot 2 > 1 \cdot 2$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x - 2}{2} \cdot 2 > 1 \cdot 2$

Step 2.2.1.1.2

Rewrite the expression.

$x - 2 > 1 \cdot 2$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $2$ by $1$ .

$x - 2 > 2$

Step 2.3

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

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Step 2.3.1

Add $2$ to both sides of the inequality.

$x > 2 + 2$

Step 2.3.2

Add $2$ and $2$ .

$x > 4$

Step 3

Solve $- \frac{x - 2}{2} > 1$ for $x$ .

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Step 3.1

Divide each term in $- \frac{x - 2}{2} > 1$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \frac{x - 2}{2} > 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 2}{2}}{- 1} < \frac{1}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 2}{2}}{1} < \frac{1}{- 1}$

Step 3.1.2.2

Divide $\frac{x - 2}{2}$ by $1$ .

$\frac{x - 2}{2} < \frac{1}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $1$ by $- 1$ .

$\frac{x - 2}{2} < - 1$

Step 3.2

Multiply both sides by $2$ .

$\frac{x - 2}{2} \cdot 2 < - 1 \cdot 2$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{x - 2}{2} \cdot 2 < - 1 \cdot 2$

Step 3.3.1.1.2

Rewrite the expression.

$x - 2 < - 1 \cdot 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 1$ by $2$ .

$x - 2 < - 2$

Step 3.4

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

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Step 3.4.1

Add $2$ to both sides of the inequality.

$x < - 2 + 2$

Step 3.4.2

Add $- 2$ and $2$ .

$x < 0$

Step 4

Find the union of the solutions.

$x < 0$ or $x > 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > 4$

Interval Notation:

$(- \infty, 0) \cup (4, \infty)$

Step 6