Algebra Examples

| x | \geq 0

$| x | \geq 0$

Step 1

Write

| x | \geq 0

$| x | \geq 0$ 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.

x \geq 0

$x \geq 0$

Step 1.2

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

x < 0

$x < 0$

Step 1.3

In the piece where

x

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

- 1

$- 1$ .

- x \geq 0

$- x \geq 0$

Step 1.4

Write as a piecewise.

{\begin{matrix} x \geq 0 & x \geq 0 - x \geq 0 & x < 0 \end{matrix}

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

{\begin{matrix} x \geq 0 & x \geq 0 - x \geq 0 & x < 0 \end{matrix}

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

Step 2

Find the intersection of

x \geq 0

$x \geq 0$ and

x \geq 0

$x \geq 0$ .

x \geq 0

$x \geq 0$

Step 3

Solve

- x \geq 0

$- x \geq 0$ when

x < 0

$x < 0$ .

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Step 3.1

Divide each term in

- x \geq 0

$- x \geq 0$ by

- 1

$- 1$ and simplify.

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Step 3.1.1

Divide each term in

- x \geq 0

$- x \geq 0$ by

- 1

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

\frac{- x}{- 1} \leq \frac{0}{- 1}

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

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{x}{1} \leq \frac{0}{- 1}

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 3.1.2.2

Divide

x

$x$ by

1

$1$ .

x \leq \frac{0}{- 1}

$x \leq \frac{0}{- 1}$

x \leq \frac{0}{- 1}

$x \leq \frac{0}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide

0

$0$ by

- 1

$- 1$ .

x \leq 0

$x \leq 0$

x \leq 0

$x \leq 0$

x \leq 0

$x \leq 0$

Step 3.2

Find the intersection of

x \leq 0

$x \leq 0$ and

x < 0

$x < 0$ .

x < 0

$x < 0$

x < 0

$x < 0$

Step 4

Find the union of the solutions.

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$

Step 6