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Chapter 1: Problem 92

Write a true sentence using either \(<,>,\) or \(=\). $$|-5| \square|-2|$$

Short Answer

Expert verified
|-5| > |-2|.

Step by step solution

01

Understand Absolute Value

Absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, |x| is always positive or zero.
02

Calculate Absolute Values

Compute the absolute values for both numbers: |-5| = 5 |-2| = 2
03

Compare the Values

Compare the absolute values calculated in the previous steps: 5 > 2
04

Write the Sentence

Now, replace the square with the correct symbol: |-5| > |-2|.

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Key Concepts

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

Absolute Value
Understanding the absolute value of a number is crucial when working with inequalities. Absolute value is represented by the symbols \(|...|\), and it signifies the distance of a number from zero on the number line. This distance is always positive or zero.
For instance:
  • \(|-5| = 5\) because the distance of -5 from zero is 5 units.
  • \(|2| = 2\) because the distance of 2 from zero is 2 units.
It's important to know that the absolute value essentially ignores whether the number is negative or positive, focusing solely on its magnitude.
Inequality Symbols
When comparing absolute values, you'll often use inequality symbols such as \(<\), \(>\), and \(=\).
These symbols help us understand how one value stands relative to another.
  • The symbol \(<\) means 'less than'.
  • The symbol \(>\) means 'greater than'.
  • The symbol \(=\) means 'equal to'.
For example, if you have \(5 > 2\), it means 5 is greater than 2. When working with absolute values, like \(|-5|\) and \(|-2|\), these symbols help you to clearly state which absolute value is larger or smaller.
In our solved example: \(|-5| > |-2|\), because 5 (the absolute value of -5) is greater than 2 (the absolute value of -2).
Number Line
A number line is a visual representation that helps us understand the position and comparison of numbers.
It is a straight line where every point corresponds to a real number, arranged in increasing order from left to right.
When comparing absolute values, a number line can be quite helpful.
For example, consider placing -5 and 2 on a number line.
  • -5 will be to the left of 0.
  • 2 will be to the right of 0.
To understand their absolute values, we consider their distance from zero, regardless of direction.
  • |-5| is the distance of -5 from 0, which is 5 units to the left.
  • |2| is the distance of 2 from 0, which is 2 units to the right.
Therefore, the absolute value comparison becomes clearer: 5 units (for -5) is greater than 2 units (for 2), so \(|-5| > |-2|\).

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