|x|>12

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

x≥0

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

x>12

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>12

Write as a piecewise.

{x>12x≥0-x>12x<0

{x>12x≥0-x>12x<0

Find the intersection of x>12 and x≥0.

x>12

Multiply each term in -x>12 by -1

Multiply each term in -x>12 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

(-x)⋅-1<12⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x<12⋅-1

Multiply x by 1.

x<12⋅-1

x<12⋅-1

Multiply 12 by -1.

x<-12

x<-12

Find the intersection of x<-12 and x<0.

x<-12

x<-12

Find the union of the solutions.

x<-12 or x>12

The result can be shown in multiple forms.

Inequality Form:

x<-12 or x>12

Interval Notation:

(-∞,-12)∪(12,∞)

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