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Chapter 8: Problem 56

Solve each inequality. Graph the solution set and write it using interval notation. \(|x|<9\)

Short Answer

Expert verified
The solution is \(-9 < x < 9\) or in interval notation: (-9, 9).

Step by step solution

01

Understand the Inequality

The inequality \(|x| < 9\) states that the absolute value of \(x\) must be less than 9. This means that \(x\) can take any value between -9 and 9, excluding -9 and 9 themselves.
02

Write the Inequality without Absolute Value

To solve \(|x| < 9\), we rewrite it as a compound inequality: -9 < x < 9. This tells us that \(x\) must lie between -9 and 9.
03

Graph the Solution on a Number Line

To graph the solution, draw a number line and place open circles at -9 and 9 because these values are not included in the solution. Shade the region between -9 and 9, representing all values \(x\) can take.
04

Write the Solution using Interval Notation

Using interval notation, the solution is written as (-9, 9). The parentheses indicate that -9 and 9 are not part of the solution set.

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Key Concepts

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

Understanding Absolute Value Inequalities
Absolute values measure the distance a number is from zero on the number line, without considering direction. In the inequality \(|x|<9\), we want the distance between the number \(x\) and zero to be less than 9. This doesn't tell us the exact value of \(x\), but rather that \(x\) is somewhere between -9 and 9, not touching either point.

To remove the absolute value, you rewrite it as a double inequality: \(-9 < x < 9\). This shows that \(x\) includes every value between these two numbers, but not the endpoints.

Remember, when solving inequalities with absolute values:
  • \(|x|
  • \(|x|>a\) transforms into two separate inequalities: \(x>a\) or \(x<-a\).
Each scenario has a different method, but understanding the core idea is crucial.
Graphing Inequality Solutions
Graphing is a great visual tool for understanding the range of values that solve an inequality. For our inequality \(-9 < x < 9\), think about a number line.

First, mark -9 and 9 with open circles, because these numbers are not part of the solution set. Open circles indicate that the endpoints are excluded. Then, shade the line in between these two points. This shaded region represents all the possible values that \(x\) could be.
  • Always use open circles for inequalities like \(<\) and \(>\).
  • Use closed circles when inequalities are \(\leq\) or \(\geq\).
Graphing transforms abstract problem-solving into something more tangible and easier to understand.
Using Interval Notation
Interval notation is a shorthand way to express the set of solutions for an inequality. It uses brackets and parentheses to describe the start and end points of the solution set.

For \(-9 < x < 9\), written in interval notation, it becomes \((-9, 9)\). The parentheses signal that -9 and 9 are not included in the solution.
  • Use parentheses \(()\) to exclude endpoints from the solution set.
  • Use brackets \([]\) to include endpoints in the solution set.
For example, if the inequality allowed \(x\) to be -9 or 9, it would be written as \([-9, 9]\) in interval notation. Understanding interval notation is important for clearly presenting solutions in a concise form.

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Most popular questions from this chapter

Customer Service. \(\quad\) A software service hotline has found that on Mondays, the polynomial function \(C(t)=-0.0625 t^{4}+t^{3}-6 t^{2}+16 t\) approximates the number of callers to the hotline at any one time. Here, \(t\) represents the time, in hours, since the hotline opened at 8: 00 A.M. How many service technicians should be on duty on Mondays at noon if the company doesn't want any callers to the hotline waiting to be helped by a technician?

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Perform the operations and simplify, if possible. See Example 6 $$\frac{3 n^{2}+5 n-2}{12 n^{2}-13 n+3} \div \frac{n^{2}+3 n+2}{4 n^{2}+5 n-6}$$

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