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Chapter 3: Problem 36

Solve and write interval notation for the solution set. Then graph the solution set. $$|x|<3$$

Short Answer

Expert verified
The solution set in interval notation is \((-3, 3)\), and it's graphed with open circles at -3 and 3, and shading in between.

Step by step solution

01

Understand the Absolute Value Inequality

An absolute value inequality such as \(|x| < 3\) means that the distance between \(x\) and 0 on the number line is less than 3.
02

Rewrite the Inequality without Absolute Value

Rewrite the absolute value inequality \(|x| < 3\) as a compound inequality. This gives us: \(-3 < x < 3\).
03

Write the Interval Notation

The inequality \(-3 < x < 3\) can be expressed in interval notation as \((-3, 3)\).
04

Graph the Solution Set

On a number line, graph the interval \((-3, 3)\) by making an open circle at -3 and 3, then shading the region in between these points to indicate all the values \(x\) can take.

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Key Concepts

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

interval notation
Interval notation is a way of representing ranges of values on the number line. It is very useful for expressing solution sets of inequalities.
There are three types of brackets used in interval notation:
  • Round brackets ( ) indicate that the endpoint is not included.
  • Square brackets [ ] show that the endpoint is included.
  • Combinations such as ( ) or [ ] are used depending on whether each endpoint is included or not.

For the inequality \( -3 < x < 3 \), the endpoints -3 and 3 are not included. So, we use round brackets: \( (-3, 3) \).
This tells us that all solutions for x lie between -3 and 3 but do not actually include -3 or 3 themselves.
compound inequalities
A compound inequality involves two separate inequalities combined into one statement by the word 'and' or 'or.'
For \(|x| < 3\), think of this visually: It means that the distance of x from 0 must be less than 3.
This translates to \(-3 < x < 3\), which is our compound inequality.
Here, 'and' is implied between \(-3 < x\) and \(x < 3\) because both conditions must be satisfied simultaneously.
Compound inequalities can often be split into two separate inequalities for easier solving, but keep in mind you are describing the same range of values.
graphing inequalities
Graphing inequalities on a number line helps visualize the set of all possible solutions. Let鈥檚 graph \(-3 < x < 3\) as an example:
  • Start by drawing a number line.
  • Mark the points -3 and 3 on this line.
  • Since -3 and 3 are not included in our solution (they have open brackets), place an open circle on each point.
  • Shade the region between -3 and 3. This shaded section shows all the points x can take.

Properly graphing inequalities helps not only in verifying solution sets but also in understanding how solutions relate to one another. Beyond a single number line, these skills are foundational for graphing more complex sets of inequalities in multiple variables.

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