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Chapter 1: Problem 58

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |3 x| \geq 0 $$

Short Answer

Expert verified
The solution is \( x \in \mathbb{R} \) or \( (-\infty, \infty) \).

Step by step solution

01

Understand the Inequality

The given inequality is \( |3x| \geq 0 \). The absolute value \( |3x| \) represents the distance of \( 3x \) from zero, which is always non-negative.
02

Analyze the Absolute Value

Since the absolute value of any real number is always greater than or equal to zero, \( |3x| \geq 0 \) is always true for all real values of \( x \). This means that there are no restrictions on \( x \).
03

Express the Solution in Set Notation

Since \( x \) can be any real number, the solution set in set notation is \( x \in \mathbb{R} \).
04

Express the Solution in Interval Notation

In interval notation, the solution set is expressed as \( (-\infty, \infty) \).
05

Graph the Solution Set

Since the solution set includes all real numbers, the entire number line is shaded to represent \( x \in \mathbb{R} \).

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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 method to describe the set of all numbers between a starting and an ending point.
It is especially useful for denoting solution sets of inequalities.

In interval notation, we use brackets [ ] and parentheses ( ) to show the inclusion or exclusion of endpoints.
  • A square bracket [ or ] indicates that the endpoint is included in the interval.
  • A parenthesis ( or ) indicates that the endpoint is not included in the interval.
For example:
- \( [2, 5] \) means all numbers from 2 to 5, inclusive.
- \( (2, 5) \) means all numbers from 2 to 5, exclusive.

In the context of our exercise, since there are no restrictions on \( x \), the interval notation for the solution set is \( (-\infty, \infty) \), which means all real numbers.
Set Notation
Set notation is another way to describe a collection of elements, often numbers, that satisfy a certain condition.
It is often used in mathematics to describe solution sets for equations and inequalities.

In set notation, curly braces { } are used, and we define the set with a rule or condition.
For example:
  • \( \{ x \mid x \, \text{is an integer} \} \) means the set of all x such that x is an integer.
  • \( \{ x \in \mathbb{R} \mid x > 0 \} \) means the set of all real numbers x such that x is greater than 0.
In our absolute value inequality \( |3x| \geq 0 \), this is true for all real numbers \( x \).
Therefore, the set notation for our solution set is \( \{ x \in \mathbb{R} \} \), which means x is any real number.
Solution Set
The solution set of an inequality is the set of all values that satisfy the inequality.
For our inequality \( |3x| \geq 0 \), this means finding all x for which this statement holds true.

The absolute value of any real number is always non-negative.
Since \( |3x| \) is never less than zero, the inequality holds for all real numbers.
  • There are no specific restrictions on \( x \).
  • This makes the solution set all real numbers.
In mathematical notation, the solution set can be expressed as both:
  • Set notation: \( \{ x \in \mathbb{R} \} \)
  • Interval notation: \( (-\infty, \infty) \)
Both notations mean the same thing: all possible real number values.
Number Line
A number line graphically represents the set of all real numbers.
It is a straight line where each point corresponds to a real number.

To graph the solution set on a number line:
  • Mark any important points or intervals.
  • Shade or highlight the regions that belong to the solution set.
For our inequality \( |3x| \geq 0 \), since every real number is a solution, we need to:
  • Shade the entire number line to indicate that all real numbers \( x \) are included.
This visual representation can help reinforce the understanding that the solution set includes every possible real number.

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