Finite Math Examples

$| z | z > 4$

Step 1

Write $| z | z > 4$ 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.

$z \geq 0$

Step 1.2

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

$z \cdot z > 4$

Step 1.3

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

$z < 0$

Step 1.4

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

$- z \cdot z > 4$

Step 1.5

Write as a piecewise.

${\begin{cases} z \cdot z > 4 & z \geq 0 \\ - z \cdot z > 4 & z < 0 \end{cases}$

Step 1.6

Multiply $z$ by $z$ .

${\begin{cases} z^{2} > 4 & z \geq 0 \\ - z \cdot z > 4 & z < 0 \end{cases}$

Step 1.7

Multiply $z$ by $z$ by adding the exponents.

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Step 1.7.1

Move $z$ .

${\begin{cases} z^{2} > 4 & z \geq 0 \\ - (z \cdot z) > 4 & z < 0 \end{cases}$

Step 1.7.2

Multiply $z$ by $z$ .

${\begin{cases} z^{2} > 4 & z \geq 0 \\ - z^{2} > 4 & z < 0 \end{cases}$

Step 2

Solve $z^{2} > 4$ when $z \geq 0$ .

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Step 2.1

Solve $z^{2} > 4$ for $z$ .

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Step 2.1.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{z^{2}} > \sqrt{4}$

Step 2.1.2

Simplify the equation.

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Step 2.1.2.1

Simplify the left side.

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Step 2.1.2.1.1

Pull terms out from under the radical.

$| z | > \sqrt{4}$

Step 2.1.2.2

Simplify the right side.

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Step 2.1.2.2.1

Simplify $\sqrt{4}$ .

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Step 2.1.2.2.1.1

Rewrite $4$ as $2^{2}$ .

$| z | > \sqrt{2^{2}}$

Step 2.1.2.2.1.2

Pull terms out from under the radical.

$| z | > | 2 |$

Step 2.1.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| z | > 2$

Step 2.1.3

Write $| z | > 2$ as a piecewise.

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Step 2.1.3.1

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

$z \geq 0$

Step 2.1.3.2

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

$z > 2$

Step 2.1.3.3

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

$z < 0$

Step 2.1.3.4

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

$- z > 2$

Step 2.1.3.5

Write as a piecewise.

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

Step 2.1.4

Find the intersection of $z > 2$ and $z \geq 0$ .

$z > 2$

Step 2.1.5

Divide each term in $- z > 2$ by $- 1$ and simplify.

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Step 2.1.5.1

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

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

Step 2.1.5.2

Simplify the left side.

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Step 2.1.5.2.1

Dividing two negative values results in a positive value.

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

Step 2.1.5.2.2

Divide $z$ by $1$ .

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

Step 2.1.5.3

Simplify the right side.

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Step 2.1.5.3.1

Divide $2$ by $- 1$ .

$z < - 2$

Step 2.1.6

Find the union of the solutions.

$z < - 2$ or $z > 2$

Step 2.2

Find the intersection of $z < - 2 or z > 2$ and $z \geq 0$ .

$z > 2$

Step 3

Solve $- z^{2} > 4$ for $z$ .

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Step 3.1

Divide each term in $- z^{2} > 4$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- z^{2} > 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- z^{2}}{- 1} < \frac{4}{- 1}$

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.

$\frac{z^{2}}{1} < \frac{4}{- 1}$

Step 3.1.2.2

Divide $z^{2}$ by $1$ .

$z^{2} < \frac{4}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $4$ by $- 1$ .

$z^{2} < - 4$

Step 3.2

Since the left side has an even power, it is always positive for all real numbers.

No solution

Step 4

Find the union of the solutions.

$z > 2$

Step 5

Convert the inequality to interval notation.

$(2, \infty)$

Step 6