Problem 77 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=\) $$ -6 ? 2 $$

Step-by-Step Solution

Answer

-6 < 2

Step 1 - Understand numerical positions on number line

First, one should understand the basic numerical placement rule on the number line. In number line, a number on the right of the other number is greater. As here, 2 is on the right to -6, so 2 is greater than -6.

Step 2 - Apply rule for numerical comparison

Now apply the numerical comparison rule. If a number is greater than the other number, the other number is less than this number. Since 2 is greater than -6, -6 is less than 2.

Key Concepts

Understanding the Number Line

The number line is a visual representation of all real numbers in order. It's like a map where every point on the line corresponds to a real number. Here are some important features:

• The numbers increase as you move to the right.
• The numbers decrease as you move to the left.
• Zero (\( 0 \)) is the center, with positive numbers on the right and negative numbers on the left.

For example, if you were to stand at zero and observe -6 and 2 on a number line, you'd see that 2 is on the right of -6. This visual can help us understand that 2 is greater than -6. Visual aids like this are vital for understanding how numbers relate to each other. By simply noting their positions relative to zero and to each other, you can quickly ascertain which numbers are larger or smaller.

Numerical Comparison with a Number Line

Numerical comparison is like a game of size. By using a number line, comparing two numbers becomes straightforward. Here's how:

• Locate each number on the number line.
• Determine their positions relative to each other.
• The number further to the right is always greater.

In the exercise, -6 is clearly to the left of 2. Hence, when comparing these two, observe that:

• The position of -6 tells us it is less than 2.
• You can make this determination swiftly by remembering that numbers to the right are larger.

Practicing this with different pairs of numbers on a number line helps to develop strong skills in numerical comparison. The number line acts as a simple yet powerful tool to visualize and compare "greater than" or "less than" relationships.

Greater Than and Less Than Symbols

In mathematics, symbols are as crucial as words in a language. The symbols \( > \) and \( < \) are used to compare two numbers. You can think of them as the "greater than" and "less than" signs.

• If one number is larger, use \( > \) (greater than).
• If one number is smaller, use \( < \) (less than).
• If the numbers are equal, the \( = \) sign is used.

Applying these in our exercise, since 2 is greater than -6, we write:\[-6 < 2\]Remembering this is key to solving many math problems. The pointy side of the symbol always faces the smaller number. Practicing with examples will help solidify your understanding of these crucial concepts.