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Chapter 2: Problem 45

Solve each inequality. Graph the solution set and write it in interval notation. $$ |x|>0 $$

Short Answer

Expert verified
The solution set is \((-\infty, 0) \cup (0, \infty)\), graphing all real numbers except zero.

Step by step solution

01

Understand the Absolute Value Inequality

An absolute value inequality like \(|x| > 0\) implies that the value of \(x\) must be something other than zero, since absolute value refers to the distance from 0, which is always non-negative.
02

Break Down the Inequality

The inequality \(|x| > 0\) means that either \(x > 0\) or \(x < 0\). Absolute value inequalities have this logic since they represent distance in both positive and negative directions from 0.
03

Analyze Both Cases

Since \(x > 0\) and \(x < 0\) comprise all non-zero real numbers, the solution set for \(|x| > 0\) is all \(x\) except \(0\). Thus, it can be written as \(x \in (-\infty, 0) \cup (0, \infty)\).
04

Graph the Solution Set

To graph the solution set of \(|x| > 0\), draw a number line and shade all regions except \(x = 0\). Place open circles at 0 to indicate that \(x = 0\) is not included in the solution.
05

Write in Interval Notation

The solution in interval notation is \((-\infty, 0) \cup (0, \infty)\), reflecting that \(x\) can be any number except 0.

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Key Concepts

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

Solution Set
When working with absolute value inequalities, understanding the solution set is key. For the inequality \(|x| > 0\), we need to identify all values of \(x\) that make the sentence true. The absolute value of a number represents its distance from zero on a number line and is always non-negative.
Since we are looking for values where \(|x| > 0\), \(x\) must be anything other than zero. This means our solutions are all real numbers except zero.
To express this in terms of a solution set:
  • If \(x > 0\), then \(x\) has positive values.
  • If \(x < 0\), then \(x\) has negative values.
Combining these solutions, the complete solution set includes all numbers less than zero and more than zero, essentially all real numbers, excluding exactly zero.
Interval Notation
Interval notation is a convenient way to denote the solution set of an inequality. It uses intervals to show where solutions exist on the number line. For the absolute value inequality \(|x| > 0\), we need to express all numbers except zero.
We distinguish the intervals where the solutions lie:
  • The solution for \(x < 0\), would be expressed as \((-\infty, 0)\).
  • For \(x > 0\), the solution is \((0, \infty)\).
Putting these together, the interval notation becomes \((-\infty, 0) \cup (0, \infty)\). The union symbol \(\cup\) signifies that the solution consists of values from both these intervals. This expresses that all numbers except for zero are included in our solution set.
Number Line Graphing
Graphing the solution set of an inequality on a number line provides a visual representation of the values \(x\) can take. For \(|x| > 0\), we consider every real number except zero.
To graph \(|x| > 0\):
  • Draw a straight line to represent the number line.
  • Mark an open circle at 0 to show that 0 is not included in the solution set.
  • Shade the entire line to the left of 0 (for negative values) and to the right of 0 (for positive values) indicating that all numbers other than 0 are solutions.
This visual indicates to the viewer that all numbers, apart from zero, satisfy the inequality, reinforcing the solution set we established in previous steps.

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Most popular questions from this chapter

Solve each compound inequality. Write solutions in interval notation. See Examples 2 and \(3 .\) $$ 2 x+4>0 \text { and } 4 x>0 $$

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