Convert to Interval Notation |x|>1

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Algebra

$| x | > 1$

Write $| x | > 1$ as a piecewise.

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

$x \geq 0$

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

$x > 1$

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

$x < 0$

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

$- x > 1$

Write as a piecewise.

${\begin{cases} x > 1 & x \geq 0 \\ - x > 1 & x < 0 \end{cases}$

Find the intersection of $x > 1$ and $x \geq 0$ .

$x > 1$

Solve $- x > 1$ when $x < 0$ .

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Multiply each term in $- x > 1$ by $- 1$

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

$(- x) \cdot - 1 < 1 \cdot - 1$

Multiply $(- x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x < 1 \cdot - 1$

Multiply $x$ by $1$ .

$x < 1 \cdot - 1$

Multiply $- 1$ by $1$ .

$x < - 1$

Find the intersection of $x < - 1$ and $x < 0$ .

$x < - 1$

Find the union of the solutions.

$x < - 1$ or $x > 1$

Convert the inequality to interval notation.

$(- \infty, - 1) \cup (1, \infty)$

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