Question

Express the inequality 2 < |x - 1/2| < 3 in set notation form.

Answer 1

The inequality 2 < |x - 1/2| < 3 in set notation is represented as x < -3/2 or x > 5/2, signifying that x is part of the set if x is either less than -3/2 or greater than 5/2.

Step-by-step explanation:

To express the inequality 2 < |x - 1/2| < 3 in set notation form, we need to consider two cases because of the absolute value. The absolute value of a number x is the distance between x and zero on the number line, which is always non-negative.

First, if x - 1/2 is positive, then we drop the absolute bars and solve the inequality:

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• 2 < x - 1/2
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• 2 + 1/2 < x
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• 5/2 < x

Next, if x - 1/2 is negative, we take the opposite of the inequality inside the absolute value:

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• 2 < -(x - 1/2)
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• 2 < -x + 1/2
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• 2 - 1/2 < -x
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• 3/2 < -x
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• x < -3/2

To combine these two cases, we take the union of the solutions:

Set Notation: \{x | x < -3/2 \text{ or } x > 5/2\}

This notation indicates the set of all x such that x is less than -3/2 or x is greater than 5/2, not including -3/2 and 5/2 themselves. So, the solution set includes all real numbers smaller than -3/2 and all numbers larger than 5/2.