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Chapter 2: Problem 29

Replace each with \(\langle,>,\) or \(=\) to make a true sentence. $$ -6 __-2 $$

Short Answer

Expert verified
-6 < -2

Step by step solution

01

Understand the comparison symbols

We need to replace the blank with either \(<\), \(>\), or \(=\). The symbol \(<\) means 'less than', \(>\) means 'greater than', and \(=\) means 'equal to'.
02

Analyze the given numbers

We have two numbers: -6 and -2. Let's consider their positions on the number line. A larger negative number (like -6) is to the left of a smaller negative number (like -2) on the number line.
03

Perform the comparison

Since -6 is to the left of -2 on the number line, -6 is less than -2. Therefore, we replace the blank with \(<\).

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Key Concepts

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

Comparing Numbers
Understanding how to compare numbers is a fundamental prealgebra skill. The symbols we use:
  • <: 'less than'
  • >: 'greater than'
  • =: 'equal to'
help us describe the relationship between two numbers.
For instance, if you have numbers such as -6 and -2, the question is to determine which one is larger or smaller.
To make this comparison, think about which number appears first when you count upwards from smaller to larger numbers.
This is especially important when dealing with negative numbers where common intuitions might be deceiving.
Number Line
The number line is a very useful tool in math, especially when dealing with integers such as positive and negative numbers.
Picture it as a straight horizontal line that extends infinitely in both directions.
Zero is your central point on this line; numbers to the right of zero are positive, and those to the left are negative.
To compare numbers using a number line, first locate both numbers on it.
Numbers increase in value as you move from left to right; hence, a number further to the right is always greater.
For our example, -6 and -2, -6 is positioned to the left of -2 which means -6 is less than -2.
Negative Numbers
Negative numbers can be tricky because they represent values less than zero. Imagine temperatures dropping below zero in winter; that's the concept here.
Unlike positive numbers where moving further right increases the value, for negative numbers, moving further left means you have a lower value.
  • -1 is greater than -3 because it is closer to zero
  • -6 is less than -2 because it is further away from zero
Remember this rule: The smaller the negative number is, the larger its value is on the number line.
Understanding this concept will help you correctly fill in those comparison symbols like \(<\), \(>\), and \(=\).

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Most popular questions from this chapter

In the newspaper, Ruben read that the low temperature for the day was expected to be \(-5^{\circ} \mathrm{F}\) and the high temperature was expected to be \(8^{\circ} \mathrm{F}\). What was the difference in the expected high and low temperatures?

Evaluate each expression. \(9 s,\) if \(s=-11\)

Evaluate each expression \(|7|\)

A driver depresses the brake pedal of her car and begins decelerating at a rate of 2.3 meters per second per second. How much will the car's speed change if the brake is applied for 6 seconds?

Evaluate each expression. \(-7 n,\) if \(n=-4\)
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