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Chapter 5: Problem 43

Insert \(<,>,\) or \(=\) to form a true statement. $$ -0.0932 \quad-0.0923 $$

Short Answer

Expert verified
-0.0932 < -0.0923

Step by step solution

01

Understand the Problem

We need to compare two negative decimal numbers: \(-0.0932\) and \(-0.0923\), and insert the correct inequality sign (\(<\), \(>\), or \(=\)) to make the statement true.
02

Compare Absolute Values

Remove the negative signs temporarily and compare the absolute values: \(0.0932\) and \(0.0923\). \(0.0932\) is greater than \(0.0923\).
03

Apply Negative Sign Rule

Since \(0.0932\) is greater than \(0.0923\), the negative of this would mean \(-0.0932\) is less than \(-0.0923\). In negative numbers, larger absolute values correspond to smaller actual values.
04

Write the Final Inequality

The true statement is \(-0.0932 < -0.0923\).

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Key Concepts

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

Absolute Values
Absolute values help us understand the magnitude of a number without considering its sign. This is particularly useful when dealing with negative numbers, as it provides a way to compare their sizes directly. The absolute value of a number is always non-negative, even if the original number is negative.
  • For example, the absolute value of \(-0.0932\) is \(0.0932\).
  • The absolute value of \(-0.0923\) is \(0.0923\).
By removing the negative signs, we simplify the process of comparing two numbers. We only consider their size or distance from zero, without the complication of dealing with the direction or sign. Thus, comparing \(0.0932\) and \(0.0923\) is straightforward, and we see that \(0.0932\) is indeed larger.
Inequality Symbols
Inequality symbols help compare the relative sizes of two numbers. The three main inequality symbols are:
  • \(<\) means "less than."
  • \(>\) means "greater than."
  • \(=\) means "equal to."
When comparing numbers, the choice of inequality symbol reflects the relationship between these numbers. For example, using \(-0.0932\) and \(-0.0923\) as our example:- Since \(-0.0932\) is less than \(-0.0923\), we use the symbol \(<\).Understanding inequality symbols is important because it allows us to accurately communicate mathematical relationships. They serve as vital tools in both mathematics and real-world problem-solving.
Negative Numbers Comparison
Comparing negative numbers might seem tricky at first, but it follows its own logical rules. Negative numbers get smaller as their absolute values increase. This is the opposite of positive numbers.- For example, \(-0.0932\) is further from zero than \(-0.0923\). This means that \(-0.0932\) is actually less than \(-0.0923\).Here are a few key points to remember when comparing negative numbers:
  • If two numbers are negative, the number with the larger absolute value is actually smaller.
  • Understanding the placement on the number line helps visualize this鈥攏umbers further left are smaller.
  • Always convert to absolute values first when comparing, then reintroduce the signs to understand their true relationship.
With these rules, comparing negative numbers becomes straightforward, offering clarity and simplicity even in seemingly complex comparisons.

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