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Chapter 0: Problem 44

Write each set as an interval or of two intervals. $$ \\{x:|x|>9\\} $$

Short Answer

Expert verified
The set \(\{x: |x| > 9\}\) can be represented as the interval \((-\infty, -9) \cup (9, \infty)\).

Step by step solution

01

Understanding the inequality

We are given the inequality \(|x|>9\). This is an absolute value inequality, which means that it involves the distance of \(x\) from zero. The inequality states that the distance of \(x\) from zero is greater than 9.
02

Converting the absolute value inequality to a compound inequality

To find the interval(s) for the given inequality, we need to convert the absolute value inequality into a compound inequality. An absolute value inequality can be represented as a combination of two inequalities. For \(|x|>9\), it means that \(x\) can be less than -9 or greater than 9. Hence, we can represent it as two inequalities joined by the word "or": $$ x < -9 \:\text{or}\: x > 9 $$
03

Writing the solution as an interval or a union of intervals

Now we can express the solution as an interval or a union of intervals. For \(x < -9\), it means that \(x\) can take any value less than -9. We can express this as an interval: $$ (-\infty, -9) $$ For \(x > 9\), it means that \(x\) can take any value greater than 9. We can express this as an interval: $$ (9, \infty) $$ Since the solution includes values from both these intervals, we can write the final solution as a union of these two intervals: $$ (-\infty, -9) \cup (9, \infty) $$ The set \(\{x: |x| > 9\}\) can be represented as the interval \((-\infty, -9) \cup (9, \infty)\).

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Key Concepts

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

Absolute Value
Absolute value is a mathematical concept that measures the distance of a number from zero on a number line.
It is always non-negative because distance cannot be negative. For example, the absolute value of \[ - 3 \] is 3, written as \[ | - 3| = 3 \].The key point to understand is that absolute value strips away the sign of a number, focusing only on magnitude. This concept is crucial in many areas of mathematics, including solving inequalities, calculus, and complex numbers.
Compound Inequality
A compound inequality involves two separate inequalities joined together by either "and" or "or". When solving compound inequalities, we find solutions that satisfy either both conditions (in the case of "and") or at least one condition (in the case of "or").
For example, \[ x < -9 \] and \[ x > 9 \] come from breaking down the absolute value inequality \[ |x| > 9 \].This results in a compound inequality because the solution involves values that satisfy \[ x < -9 \] or \[ x > 9 \]. Compound inequalities help express more complex conditions compared to simple inequalities.
Union of Intervals
The union of intervals is used to show that more than one interval is part of the solution set. It is represented by the symbol "\( \cup \)", which stands for "or" in the language of inequalities.
For example, if \[ x < -9 \] results in the interval \[ (-\infty, -9) \] and \[ x > 9 \] results in the interval \[ (9, \infty) \], the solution is the union of these: \[ (-\infty, -9) \cup (9, \infty) \].This means that any number less than -9 or greater than 9 satisfies the original inequality \[ |x| > 9 \].Union of intervals is an essential tool in mathematics for expressing solutions to inequalities and set relationships.

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Most popular questions from this chapter

The intersection of two sets of numbers consists bers that are in both sets. If \(A\) and \(B\) are sets, intersection is denoted by \(A \cap B\). Write each intersection as a single interval. $$ [2,7) \cap[5,20) $$

Find all numbers \(x\) satisfying the given equation. $$ |x+1|+|x-2|=7 $$

Find all numbers with absolute value \(9 .\)

Give four examples of pairs of real numbers \(a\) and \(b\) such that \(|a+b|=2\) and \(|a|+|b|=8\).

Give an example of an open interval and a closed interval whose union equals the interval [-3,7]
See all solutions

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