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Chapter 1: Problem 24

Determine whether the following statements are true or false. $$-3<-2$$

Short Answer

Expert verified
The statement \(-3<-2\) is True.

Step by step solution

01

Understanding the concept of inequality

In mathematics, the inequality symbol '<' shows that one value is lesser than the other. In other words, if 'a' and 'b' are two numbers and \(a<b\), that means 'a' is lesser than 'b'.
02

Analyzing the provided inequality

Here, the provided inequality is \(-3<-2\), which implies, is '-3' lesser than '-2'? This can be verified by placing these two numbers on the number line.
03

Verification

In the number line, a negative number is always considered lesser than a positive number. And also, among the negative numbers, a number is considered greater if its absolute value is lesser. In this case, '-2' has a lesser absolute value than '-3', so '-2' is greater than '-3'.
04

Final Verification

Hence after observing the number line, it can be seen that '-3' is indeed lesser than '-2', or in other words, the inequality \(-3<-2\) holds true.

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Key Concepts

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

Understanding the Number Line
A number line is a simple representation where numbers are placed in an order from left to right. It helps us visualize the size and order of both positive and negative numbers.
On a number line:
  • Positive numbers are to the right of zero.
  • Negative numbers are to the left of zero.
  • The further you move to the right, the larger the number becomes.
  • The further you move to the left, the smaller (or more negative) the number appears.
By using a number line, it's easy to compare numbers to see which is larger or smaller. This is crucial when solving inequalities, like determining whether a number is greater or lesser than another.
Exploring Negative Numbers
Negative numbers might seem tricky at first, but they're simply numbers that are less than zero. They often represent values like debts or temperatures below freezing.
When working with negative numbers:
  • The larger the negative number appears, the smaller its value actually is on the number line.
  • A negative number is always less than zero and any positive number.
  • In the context of inequalities, a negative number is less than another if it is further to the left on the number line.
As an example, the inequality \(-3 < -2\) means \-\3\ is to the left of \(-2\) on the number line, making it smaller.
Interpreting Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always expressed as a non-negative number.
Here's what you should know about absolute value:
  • Denoted by vertical bars, like \(|-3| = 3|\).
  • It focuses on the size of the number, not its sign.
  • Both \(-3\) and \(+3\) have absolute values of \(3\).
Understanding absolute value can help clarify relationships between negative numbers, like in the given inequality. In \(-3 < -2\), noting that \(3 > 2\) demonstrates that \(-3\) is smaller because its absolute value is larger.

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Most popular questions from this chapter

Write each sentence as a mathematical expression. The sum of adding three and four is equal to seven.

A rectangular parking lot is 253.5 feet long and 178.5 feet wide. A 55 -gallon drum of asphalt sealer covers 4,000 square feet and costs \(\$ 97.50 .\) Find the cost to seal the parking lot. (Sealer can be purchased only in full drums.)

Use the distributive property to write each expression without parentheses. $$-6(a+4)$$

Which property of real numbers justifies each statement? $$3+(-3)=0$$

Write each inequality as an equivalent inequality in which the inequality symbol points in the opposite direction. $$3+8>8$$
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