Problem 36 - Precalculus Mathematics for Calculus

Question

State whether each inequality is true or false. (a) \(1.1>1 . \overline{1}\) (b) \(8 \leq 8\)

Step-by-Step Solution

Answer

(a) False, (b) True.

Step 1: Understanding the Problem

We need to determine whether each inequality statement is true or false by comparing the given numbers.

Step 1: Analyzing (a)

Evaluate the inequality: \(1.1 > 1. \overline{1} \).To compare, recognize that \(1. \overline{1} \) is the repeating decimal 1.1111...Since 1.1 can be written as 1.1000, it is clear that 1.1 is less than 1.1111...Thus, \(1.1 > 1. \overline{1} \) is false.

Step 2: Analyzing (b)

Evaluate the inequality: \(8 \leq 8\).The symbol \(\leq\) means "less than or equal to."Since 8 is equal to 8, this statement satisfies the inequality.Thus, \(8 \leq 8\) is true.

Key Concepts

Understanding Repeating Decimals

A repeating decimal is a decimal number that has digits repeating in a pattern infinitely. It's crucial for comparing numbers because it tells us the exact value of seemingly complex decimals.
When you see a number like \(1.\overline{1}\), it means that the digit '1' repeats forever. Thus, it can be expressed as \(1.1111\ldots\).
To compare it with a non-repeating decimal like \(1.1\), convert \(1.1\) to \(1.1000\ldots\) to see how the two differ. This conversion reveals that \(1.1111\ldots\) is greater than \(1.1000\).
The art of understanding repeating decimals is critical in accurately comparing numbers and their values in mathematics.

Comparing Numbers

Comparing numbers involves determining which of two or more numbers is greater or smaller, or if they are equal. This skill is foundational for understanding inequalities.
When comparing decimals, it's useful to write them so they have the same number of digits after the decimal point. This makes it easier to see the differences between them. For instance, comparing \(1.1\) and \(1.\overline{1}\) involves recognizing that \(1.1\) can be seen as \(1.1000\ldots\), while \(1.\overline{1}\) is \(1.1111\ldots\).
Through this comparison, it's clear that after the decimal point, the repeating '1's in \(1.\overline{1}\) make it larger than the \(1.1000\ldots\). Therefore, in an inequality \(1.1 \gt 1.\overline{1}\), the statement would be false because \(1.1\) is actually less.

Using Inequality Symbols

Inequality symbols are tools that help us express the relationship between two numbers. The main symbols used are:

• \(>\): greater than
• \(<\): less than
• \(\geq\): greater than or equal to
• \(\leq\): less than or equal to

To apply these symbols, we need to understand the values of the numbers we are comparing. For example, \(8 \leq 8\) makes use of the 'less than or equal to' symbol.
This means "8 is less than or equal to 8," which is true since 8 is equal to 8. Understanding and using inequality symbols correctly allows us to effectively communicate mathematical concepts and relationships.