Algebra Examples

$\frac{| 2 x |}{- 6} > - 2$

Step 1

Multiply both sides by $- 6$ .

$\frac{| 2 x |}{- 6} \cdot - 6 < - 2 \cdot - 6$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{| 2 x |}{- 6} \cdot - 6$ .

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Step 2.1.1.1

Remove non-negative terms from the absolute value.

$\frac{2 | x |}{- 6} \cdot - 6 < - 2 \cdot - 6$

Step 2.1.1.2

Cancel the common factor of $6$ .

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Step 2.1.1.2.1

Factor $6$ out of $- 6$ .

$\frac{2 | x |}{6 (- 1)} \cdot - 6 < - 2 \cdot - 6$

Step 2.1.1.2.2

Factor $6$ out of $- 6$ .

$\frac{2 | x |}{6 \cdot - 1} \cdot (6 \cdot - 1) < - 2 \cdot - 6$

Step 2.1.1.2.3

Cancel the common factor.

$\frac{2 | x |}{6 \cdot - 1} \cdot (6 \cdot - 1) < - 2 \cdot - 6$

Step 2.1.1.2.4

Rewrite the expression.

$\frac{2 | x |}{- 1} \cdot - 1 < - 2 \cdot - 6$

Step 2.1.1.3

Combine $\frac{2 | x |}{- 1}$ and $- 1$ .

$\frac{2 | x | \cdot - 1}{- 1} < - 2 \cdot - 6$

Step 2.1.1.4

Simplify the expression.

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Step 2.1.1.4.1

Multiply $- 1$ by $2$ .

$\frac{- 2 | x |}{- 1} < - 2 \cdot - 6$

Step 2.1.1.4.2

Move the negative one from the denominator of $\frac{- 2 | x |}{- 1}$ .

$- 1 \cdot (- 2 | x |) < - 2 \cdot - 6$

Step 2.1.1.4.3

Rewrite $- 1 \cdot (- 2 | x |)$ as $- (- 2 | x |)$ .

$- (- 2 | x |) < - 2 \cdot - 6$

Step 2.1.1.4.4

Multiply $- 2$ by $- 1$ .

$2 | x | < - 2 \cdot - 6$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 2$ by $- 6$ .

$2 | x | < 12$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $2 | x | < 12$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 | x | < 12$ by $2$ .

$\frac{2 | x |}{2} < \frac{12}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 | x |}{2} < \frac{12}{2}$

Step 3.1.2.1.2

Divide $| x |$ by $1$ .

$| x | < \frac{12}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $12$ by $2$ .

$| x | < 6$

Step 3.2

Write $| x | < 6$ as a piecewise.

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Step 3.2.1

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

$x \geq 0$

Step 3.2.2

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

$x < 6$

Step 3.2.3

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

$x < 0$

Step 3.2.4

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

$- x < 6$

Step 3.2.5

Write as a piecewise.

${\begin{cases} x < 6 & x \geq 0 \\ - x < 6 & x < 0 \end{cases}$

Step 3.3

Find the intersection of $x < 6$ and $x \geq 0$ .

$0 \leq x < 6$

Step 3.4

Solve $- x < 6$ when $x < 0$ .

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Step 3.4.1

Divide each term in $- x < 6$ by $- 1$ and simplify.

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Step 3.4.1.1

Divide each term in $- x < 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{6}{- 1}$

Step 3.4.1.2

Simplify the left side.

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Step 3.4.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{6}{- 1}$

Step 3.4.1.2.2

Divide $x$ by $1$ .

$x > \frac{6}{- 1}$

Step 3.4.1.3

Simplify the right side.

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Step 3.4.1.3.1

Divide $6$ by $- 1$ .

$x > - 6$

Step 3.4.2

Find the intersection of $x > - 6$ and $x < 0$ .

$- 6 < x < 0$

Step 3.5

Find the union of the solutions.

$- 6 < x < 6$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 6$

Interval Notation:

$(- 6, 6)$

Step 5