Problem 48 - Elementary Algebra

Question

Fill in the blank with \(<,=,\) or \(\geqslant\). -1 _____ -200

Step-by-Step Solution

Answer

-1 ≥ -200

Step 1: Understand the Problem

We need to compare two numbers: -1 and -200. Our task is to determine which of the three symbols (<, =, ≥) correctly represents the relationship between -1 and -200.

Step 2: Identify the Numbers on the Number Line

Visualize the number line where smaller numbers are to the left, and larger numbers are to the right. -200 is positioned to the left of -1 on the number line because it is a smaller value.

Step 3: Determine the Relation

Since -1 is to the right of -200 on the number line, -1 is greater than -200. The correct symbol to use that indicates this relationship is >.

Step 4: Fill in the Blank

In the context of the options provided (which are <, =, and ≥), using established conventions, -1 is greater than -200, so the correct symbol to insert is ≥ because -1 is not less than -200, and it is definitely greater.

Key Concepts

Number Line

The number line is a visual tool that helps us understand the size and order of numbers. Imagine a horizontal line with numbers placed at equal intervals. On this line, all numbers increase as we move to the right.
The more to the right a number is, the greater its value. For example, if you place -200 and -1 on a number line:

• -200 will be more to the left since it is smaller.
• -1 will be more to the right since it is larger than -200.

By seeing where numbers fall on the number line, you can easily compare their sizes without performing any calculations. This makes it a very powerful tool for understanding relationships between numbers, especially when dealing with negatives.

Comparing Integers

Comparing integers means figuring out which number is larger or smaller. When you compare two or more integers, you want to see which one is further right or left on the number line. If a number is further to the right, it is larger.
Here are some key points when comparing integers:

• Negatives are always smaller than positives. For instance, -1 is smaller than 2.
• More negative numbers (such as -200) are smaller than less negative numbers (like -1).
• If two numbers are exactly at the same position on the number line, they are equal.

Understanding the position of integers relative to each other helps you determine their order and how they relate.

Mathematical Symbols

Mathematical symbols are the tools we use to express the relationships between numbers. They allow us to make comparisons quickly and efficiently. Here are some important symbols used when comparing numbers:

• "<" (Less than): This symbol indicates that the number on the left is smaller than the one on the right.
• "=" (Equal to): This signifies that both numbers are the same.
• "≥" (Greater than or equal to): This symbol shows that the number on the left is either greater than or exactly equal to the number on the right.

Understanding and correctly using these symbols is crucial for solving mathematical problems involving comparisons, as they help articulate which number is larger or smaller.