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Chapter 2: Problem 34

a. Which is greater, \(-14\) or \(-20 ?\) b. Which is greater, \(|-14|\) or \(|-20| ?\)

Short Answer

Expert verified
-14 is greater than -20. | -20 | is greater than | -14 |.

Step by step solution

01

Understanding Negative Numbers

Negative numbers are less than zero. The number further to the left on a number line is smaller. Therefore, compare -14 and -20 on a number line.
02

Comparing -14 and -20

-14 is closer to zero than -20. On a number line, -14 is to the right of -20, making it greater. So, -14 is greater than -20.
03

Understanding Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. It is always a non-negative number.
04

Calculating Absolute Values

Calculate the absolute values. The absolute value of -14 is | -14 | = 14. The absolute value of -20 is | -20 | = 20.
05

Comparing | -14 | and | -20 |

Compare the absolute values calculated. Since 20 is greater than 14, | -20 | is greater than | -14 |.

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They are represented with a minus sign (鈭�). For example, -14 and -20 are both negative numbers.
On a number line, negative numbers are positioned to the left of zero. The further left a number is, the smaller its value.
To visualize this, consider a number line where ... -21, -20, -19, ... -15, -14, -13, ... 0, 1, 2, and so on.
Here, -20 is further left than -14, which makes -20 smaller than -14. So, -14 is greater than -20.
Number Line
A number line is a visual representation of numbers placed in order on a straight line. It helps us to easily compare and understand the magnitude of numbers.
  • Zero is usually at the center.
  • Numbers to the right of zero are positive.
  • Numbers to the left of zero are negative.

When we compare -14 and -20 on the number line, we see that -14 is to the right of -20. This means -14 is greater than -20.
Number lines are very useful tools in mathematics because they give us a clear picture of numerical relationships.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. This value is always a non-negative number.
To find the absolute value of a number, we simply ignore its sign.
For example:
  • The absolute value of -14 is \(|-14| = 14\).
  • The absolute value of -20 is \(|-20| = 20\).
This tells us that in terms of distance from zero, -14 is 14 units away and -20 is 20 units away.
Comparing Absolute Values
When comparing absolute values, we look at the distances from zero without considering the direction (sign). In our exercise, we calculated:
The absolute value of -14 is 14.
The absolute value of -20 is 20.

Since 20 is greater than 14, the absolute value of -20 is greater than the absolute value of -14.
Remember, comparing absolute values means comparing the distances from zero. In summary, \(|-20| = 20 > |-14| = 14\).

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