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please help me with modulus inequality.

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Im assuming this is real numbers?


If a number has modulus less than or equal to 1, then that quantity must be between -1 and 1.


If a number is between -1 and 1, then it is both greater than -1 and at the same time less than +1. The set of numbers between -1 and 1 is the intersection of two sets: all numbers greater than -1 intersected with all numbers less than +1.


So at the end you need to find the intersection of the two sets because you are trying to keep the expression in the modulus between two values.
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You are best off:

1) sketch the rational function without the modulus.

2) Apply the modulus transformation to your graph, and show the intersection points with the line y=1.

3) Identify the four intersection points as the four critical values you’ve found algebraically

4) Write inequalities to describe each region where the modulus function is below the line.

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