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Chapter 0: Problem 43

Write each set as an interval or as a union of two intervals. $$\\{x:|x|>2\\}$$

Short Answer

Expert verified
The given set \(\{x: |x|>2\}\) can be expressed as an interval or union of two intervals: \((-鈭�, -2) \cup (2, +鈭�)\).

Step by step solution

01

Identify the inequality

The given expression is: \[\\{x: |x|>2\\}\] This represents all x-values for which the absolute value is greater than 2.
02

Break down the inequality

Since the absolute value is greater than 2, we can break down the inequality into two separate inequalities: 1. x > 2 2. x < -2
03

Write the inequalities as intervals

We can now write these two inequalities as intervals: 1. x > 2 can be written as (2, +鈭�) 2. x < -2 can be written as (-鈭�, -2)
04

Write the intervals as a union

Finally, we can combine these two intervals using the union symbol to show the complete range of values for x: \[(-鈭�, -2) \cup (2, +鈭�)\] So the given set can be expressed as an interval or union of two intervals: \[\\{x: |x|>2\\} = (-鈭�, -2) \cup (2, +鈭�)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
Absolute value is a way to measure the distance of a number from zero on a number line. It's always a non-negative number. We represent the absolute value of a number \( x \) with two vertical bars: \(|x|\). For example, \(|3| = 3\) and \(|-3| = 3\). This is because both 3 and -3 are three units away from zero.

When solving inequalities involving absolute values, like \(|x| > 2\), we look for numbers whose distance from zero is more than 2. This results in two conditions: either the number is greater than 2 or less than -2. Understanding absolute value is crucial for breaking down and interpreting inequalities correctly.
Union of Intervals
A union of intervals involves combining two or more intervals to express a broader set of values. In mathematics, the union is denoted by the symbol \(\cup\).

For example, if you have two intervals, say \((-鈭�, -2)\) and \((2, +鈭�)\), taking their union means combining all values from both intervals. This implies any number less than -2 or greater than 2 is included. Using union helps provide a complete picture of all possible values that satisfy certain conditions.
  • For \( x < -2 \), the interval is \((-鈭�, -2)\).
  • For \( x > 2 \), the interval is \((2, +鈭�)\).
Exploring Set Notation
Set notation offers a way to express a collection of elements. It uses curly braces \({ }\) to enclose the numbers or conditions that define the set. For instance, \(\{x: |x|>2\}\) represents all numbers \( x \) such that their absolute value is greater than 2.

In set notation, the expression inside the braces provides a rule or condition for the elements of the set. It's a concise method for specifying ranges and conditions without listing every individual element.
Using Interval Notation
Interval notation is a shorthand way of expressing a range of values on the real number line. It uses brackets and parentheses to denote closed and open intervals, respectively:
  • \((a, b)\) means \(a < x < b\) (open interval).
  • \((a, b]\) means \(a < x \leq b\) (half-open interval).
  • \([a, b]\) means \(a \leq x \leq b\) (closed interval).
When expressing the set \(\{x: |x|>2\}\) in interval notation, we represent it as \((-鈭�, -2) \cup (2, +鈭�)\), illustrating all numbers that lie outside the interval \([-2, 2]\). This notation is efficient and widely used for solutions in calculus and algebra.

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