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Chapter 4: Problem 105

Write an equivalent inequality using absolute value. $$ -3

Short Answer

Expert verified
The equivalent inequality is |x| < 3.

Step by step solution

01

Identify the inequality

The given inequality is -3 < x < 3
02

Recognize the range

The inequality shows that the variable x is situated between -3 and 3. This means the distance of x from 0 is less than 3.
03

Express it using absolute value

The distance of x from 0 can be expressed using the absolute value notation. Therefore, -3 < x < 3 can be written as |x| < 3

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Key Concepts

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

inequalities
In mathematics, an inequality is a relation that makes a comparison between two values. Typical inequality symbols include <, >, 鈮�, and 鈮�. These symbols help express that one number is smaller, larger, or equal to another, with certain conditions.

For example, in the context of our exercise, we use the inequality -3 < x < 3 to show that the variable x lies between -3 and 3.
Understanding inequalities is crucial as it sets the foundation for solving more complex problems.
absolute value
The absolute value of a number refers to its distance from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|.

In our case, |x| < 3 means that the distance of x from 0 is less than 3. Whether x is positive or negative, when we say |x|, we are only interested in how far x is from zero without considering the direction.
This concept is handy when dealing with inequalities because it simplifies the representation of numbers within specific ranges.
interval notation
Interval notation is a way to describe a set of numbers along the number line. It shows the upper and lower boundaries of the set and the type of interval:
  • Open Interval: (a, b) where both a and b are not included.
  • Closed Interval: [a, b] where both a and b are included.
  • Half-Open Interval: One boundary is included, and the other is not, e.g., (a, b] or [a, b).
In our exercise, -3 < x < 3 can be written in interval notation as (-3, 3). This notation efficiently conveys the range of x.
distance from zero
The distance from zero is another way to express absolute value. It measures how far a number is from zero on a number line, irrespective of direction.

For instance, in the inequality -3 < x < 3, we say that the distance of x from zero is less than 3. This is written as |x| < 3.
This perspective helps in understanding absolute values because it focuses on the number's position relative to zero, which is often fundamental in solving and interpreting mathematical problems.

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