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Chapter 2: Problem 39

Which is greater 10 or \(|10| ?\)

Short Answer

Expert verified
10 and \(|10|\) are equal.

Step by step solution

01

Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of its direction. It is always non-negative. For any real number \(x\), \(|x|\) is always equal to \(x\) if \(x\) is positive or zero, and \(-x\) if \(x\) is negative.
02

Find the Absolute Value

Calculate \(|10|\). Since 10 is a positive number, \(|10| = 10\).
03

Compare the Two Values

Compare the given number 10 and the absolute value \(|10|\). Both values are 10, so they are equal.

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Key Concepts

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

absolute value
Absolute value refers to the distance of a number from zero on the number line. Distance is always measured as a non-negative quantity. Therefore, the absolute value of a number is always positive or zero.
For any number, we write its absolute value as \(|x|\).
For instance:
  • If you have a number 10, then \(|10| = 10\) because 10 is a positive number.
  • If you have a number -10, then \(|-10| = 10\), because the distance from -10 to 0 on the number line is 10.
The main idea is that the direction does not matter; only the distance does.
number line
A number line is a visual representation of numbers in order, placed at intervals along a straight line.
This visual helps us understand the position and distance of numbers more clearly.
Here are some key points about the number line:
  • Zero (0) is the center point of a number line.
  • Numbers to the right of zero are positive values (e.g., 1, 2, 3, ...).
  • Numbers to the left of zero are negative values (e.g., -1, -2, -3, ...).
Using a number line makes comparing numbers and understanding their absolute values simpler.
positive and negative numbers
Numbers can be either positive or negative, depending on their position relative to zero on the number line.

**Positive Numbers:**
  • These are numbers greater than zero.
  • They are located to the right of zero on the number line.
  • Examples: 1, 2, 10, 100, ...

**Negative Numbers:**
  • These are numbers less than zero.
  • They are located to the left of zero on the number line.
  • Examples: -1, -2, -10, -100, ...
Knowing whether a number is positive or negative helps in understanding its absolute value and in comparing it with other numbers.
comparison of numbers
Comparing numbers involves determining which number is greater, smaller, or if they are equal.
Here鈥檚 how you can compare numbers:
  • By Value: Compare the numerical value of the numbers directly.
  • By Using Absolute Values: If you are comparing absolute values, consider both numbers as non-negative.
In the example given in the exercise, you compare 10 and \(|10|\).
Since \(|10|\) is the absolute value of 10 and any positive number equals its absolute value, here both 10 and \(|10|\) are equal.
Thus, 10 is equal to its absolute value, \(|10| = 10\).

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Most popular questions from this chapter

Translate each English phrase to a mathematical expression. Then simplify. Subtract 10 from 16

Assume \(a>0\) (this means that \(a\) is positive) and \(b<0\) (this means that \(b\) is negative). Find the sign of each expression. $$|a| \div b$$

Rank the numbers from least to greatest. If \(a\) represents a negative number, then what is the sign of \(-a ?\)

Translate each English phrase to a mathematical expression. Then simplify. 27 minus 40

a. What number must be multiplied by \(-4\) to obtain \(-36 ?\) b. What number must be multiplied by \(-4\) to obtain \(36 ?\)
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