Problem 42 Use one of the symbols $<$ ... [FREE SOLUTION]

Chapter 1: Problem 42

Use one of the symbols $<$ or $>$ to make each statement true. $$-5\quad\text{__}\quad-4 $$

Short Answer

Expert verified

The correct inequality is $-5 < -4$.

Step by step solution

Understanding Negative Numbers on a Number Line

Recall that on a number line, numbers increase in value as you move from left to right. This means a number further to the left is smaller. Thus, -5 is further to the left than -4.

Comparing Magnitudes

Negative numbers have a property where numbers with a smaller absolute value are actually larger when negative. Here, |-4| = 4 is less than |-5| = 5, indicating -4 is greater than -5.

Using Symbol to Represent the Relationship

Since -5 is smaller than -4 (or -4 is greater than -5), we use the "less than" symbol ( < ) to represent this relationship. Thus: -5 < -4.

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Key Concepts

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

Understanding the Number Line

The number line is a visual representation of numbers along a straight line. It helps us easily compare and visualize the size of numbers by positioning them relative to each other. On a number line:

Numbers increase in value as you move from left to right.
Conversely, as you move from right to left, their values decrease.

When dealing with negative numbers, this concept is vital. The further left a number is placed, the smaller its value. Thus, on a number line, -5 is positioned to the left of -4. This placement effectively shows that -5 is smaller than -4.

Exploring Absolute Value

The absolute value of a number is its distance from zero on the number line, disregarding whether the number is positive or negative. It is always a non-negative number and is denoted by vertical lines:

|-4| = 4
|-5| = 5

Absolute value is crucial when comparing negative numbers. Although -4 and -5 are negative, their absolute values, which are 4 and 5 respectively, reveal a different picture. The number with the smaller absolute value (-4) is actually greater on the number line than the number with the larger absolute value (-5). This is why -4 is greater than -5, even though 4 is less than 5 when ignoring signs.

Decoding Negative Numbers

Negative numbers can sometimes be counterintuitive because they operate differently than positive numbers. Here are some important characteristics:

Negative numbers are always less than positive numbers.
The further a negative number is from zero, the smaller it is.

In real-world contexts like temperature or altitude, negative numbers are commonplace. Understanding their properties is essential for comparing them correctly. For the example -5 and -4, even though -5 appears numerically larger because of its absolute value, it's actually smaller on the number line. This is why -5 < -4. Grasping this concept helps in solving inequalities with ease.

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91影视

Use one of the symbols \(<\) or \(>\) to make each statement true. $$-5\quad\text{__}\quad-4 $$

Short Answer

Step by step solution

Understanding Negative Numbers on a Number Line

Comparing Magnitudes

Using Symbol to Represent the Relationship

Key Concepts

Understanding the Number Line

Exploring Absolute Value

Decoding Negative Numbers

One App. One Place for Learning.

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