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Chapter 1: Problem 69

Decide whether each statement is true or false. \(|-6|<|-9|\)

Short Answer

Expert verified
The statement \(|-6| < |-9|\) is true.

Step by step solution

01

Understand the Problem

We need to determine whether the statement \(|-6| < |-9|\) is true or false.
02

Recall the Definition of Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For any number \(a\), the absolute value is denoted by \(|a|\).
03

Calculate Absolute Values

Compute \(|-6|\) and \(|-9|\). \(|-6| = 6\) because \(-6\) is 6 units away from zero. \(|-9| = 9\) because \(-9\) is 9 units away from zero.
04

Compare the Absolute Values

Now, compare the calculated values: \(6 < 9\).
05

Form the Conclusion

Since \(6 < 9\), the original statement \(|-6| < |-9|\) is true.

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Key Concepts

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

absolute value
The absolute value of a number is essentially its distance from zero on the number line.
Regardless of the direction on the number line, whether the number is positive or negative, absolute value transforms it to a non-negative number.
For example, the absolute value of both 5 and -5 is 5, since both +5 and -5 are 5 units away from zero.
It is represented using vertical bars: \(|a|\).
This means \(|-6|\) is 6 and \(|-9|\) is 9.
inequality
An inequality is a mathematical statement that indicates one quantity is larger or smaller than another.
Inequalities use the symbols \(>\) for 'greater than' and \(<\) for 'less than'.
When it comes to absolute values, we compare their magnitudes.
For instance, in the exercise: we compared \(|-6|\) to \(|-9|\). Since 6 is less than 9, the inequality \(6 < 9\) holds true, making the statement \(|-6| < |-9|\) true.
number line
A number line is a visual representation of numbers on a straight line.
The center of the number line is zero, with positive numbers to the right and negative numbers to the left.
The number line helps in understanding the concept of distance from zero, which is crucial in absolute value.
For example, -6 is 6 units away from zero to the left, and -9 is 9 units away from the left.
Clearly, 6 is closer to zero than 9, which illustrates why \( |-6| \) (which equals 6) is less than \( |-9| \) (which equals 9).

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Most popular questions from this chapter

Write each sentence as an equation, using \(x\) as the variable. Then find the solution from the set of integers between \(-12\) and \(12,\) inclusive. See Example \(9 .\) The quotient of a number and 4 is \(-1\)

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Write a numerical expression for each phrase and simplify. The sum of 9 and \(-4,\) decreased by 7

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The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 3 if the sum of its digits is divisible by \(3,\) and not otherwise. Show that (a) \(4,799,232\) is divisible by 3 and \((b) 2,443,871\) is not divisible by 3
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