Precalculus Examples

$| 2 x | > 5$

Step 1

Write $| 2 x | > 5$ 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.

$2 x \geq 0$

Step 1.2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x \geq 0$ by $2$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 1.3

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

$2 x > 5$

Step 1.4

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

$2 x < 0$

Step 1.5

Divide each term in $2 x < 0$ by $2$ and simplify.

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Step 1.5.1

Divide each term in $2 x < 0$ by $2$ .

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

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.1.1

Cancel the common factor.

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

Step 1.5.2.1.2

Divide $x$ by $1$ .

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

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $2$ .

$x < 0$

Step 1.6

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

$- 2 x > 5$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 x > 5 & x \geq 0 \\ - 2 x > 5 & x < 0 \end{cases}$

Step 2

Divide each term in $2 x > 5$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x > 5$ by $2$ .

$\frac{2 x}{2} > \frac{5}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{5}{2}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x > \frac{5}{2}$

Step 3

Divide each term in $- 2 x > 5$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 x > 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{5}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{5}{- 2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x < \frac{5}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x < - \frac{5}{2}$

Step 4

Find the union of the solutions.

$x < - \frac{5}{2}$ or $x > \frac{5}{2}$

Step 5

Convert the inequality to interval notation.

$(- \infty, - \frac{5}{2}) \cup (\frac{5}{2}, \infty)$

Step 6