Problem 17 - Algebra A Combined Function

Question

Determine whether each statement is true or false. See Examples 1 through 6 and 10. $$ 5.092<5.902 $$

Step-by-Step Solution

Answer

The statement is true: 5.092 is less than 5.902.

Step 1: Understand the Inequality Sign

The inequality sign "<" means "less than." We want to determine if the number on the left is smaller than the number on the right.

Step 2: Compare Whole Number Parts

Look at the whole numbers in 5.092 and 5.902. Both have a whole number part of 5, so they are equal in this aspect.

Step 3: Compare the Tenths Place

Next, compare the digits in the tenths place. In 5.092, the tenths digit is 0. In 5.902, the tenths digit is 9. Since 0 is less than 9, 5.092 is less than 5.902.

Key Concepts

Decimal Comparison

When comparing decimal numbers, it's important to break them down step by step. Decimals are composed of a whole number part and a fractional part. To accurately compare them, we must evaluate each part separately. Start by looking at the whole number part. If the whole numbers are different, the decimal with the smaller whole number is less. If the whole numbers are the same, move to the decimal places. Focus on comparing each decimal place from left to right:

• Compare the tenths place first.
• If they are equal, compare the hundredths place next.
• Continue this process until you find a difference.

The first place where the numbers differ will tell you which number is smaller. This thorough comparison ensures accurate results.

Less Than Sign

The less than sign, denoted as "<", is a basic mathematical symbol used in inequalities. It indicates that the quantity on its left is smaller than the quantity on its right. This symbol is crucial for expressing comparisons in mathematics. To use the less than sign correctly:

• Read it as "is less than." For example, in the expression 5.092 < 5.902, it reads as "5.092 is less than 5.902."
• Ensure the numbers being compared are ordered appropriately, with the smaller number on the left side.
• The symbol always points towards the smaller number, resembling the open end of a Pac-Man figure, which "eats" the larger number.

Understanding how to use the less than sign is fundamental for solving inequalities and comparing different numbers.

Number Comparison

Number comparison is a fundamental skill in mathematics that allows us to determine the relative size of numbers. To effectively compare numbers, we often rely on place value. This involves assessing the significance of each digit depending on its position in the number. Here's a simple strategy to compare numbers:

• Start with the highest place value digit (usually the leftmost).
• Work your way to the right, comparing digits in corresponding positions.
• The first position where the numbers differ will determine which number is larger or smaller.

Number comparison is not just limited to decimals. It applies to whole numbers, fractions, and even more complex numbers. This foundational concept can assist in understanding more complex mathematical reasoning.

Mathematical Reasoning

Mathematical reasoning involves using logic to solve problems and make sense of mathematical concepts. It requires understanding rules, relationships, and patterns. In the context of comparing numbers, reasoning helps us justify and explain why one number is greater or less than another. Here are some key points about mathematical reasoning:

• It includes identifying patterns and relationships between numbers.
• Using logical steps, like those outlined in a step-by-step comparison, helps reinforce reasoning skills.
• It allows us to construct arguments and proofs to validate mathematical statements.

Developing strong mathematical reasoning skills leads to better problem-solving abilities, helping you tackle not only inequalities but also more challenging mathematical tasks with confidence.