Problem 19 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$-6 \quad 0$$

Step-by-Step Solution

Answer

-6 < 0

Step 1: Understanding Negative Numbers

Negative numbers are less than zero. On a number line, negative numbers appear to the left of zero. The further left a number is, the smaller its value.

Step 2: Comparing Negative Number to Zero

Since -6 is a negative number and zero is not, -6 will always be less than zero. This is because negative numbers are always to the left of zero on the number line.

Step 3: Placing the Correct Symbol

To express that -6 is less than 0, we need to use the symbol <. The statement will be -6 < 0.

Key Concepts

Understanding Negative Numbers

Negative numbers can be a bit tricky when you first learn about them, but they are super important for understanding how numbers interact. A negative number is any number less than zero.
Generally, these numbers are used to indicate a decrease in value or a deficiency of some kind.

• In the real world, negative numbers can represent things like temperature below zero, debts, or elevations under sea level.
• On paper, they are often presented with a minus sign in front of them, like \(-1, -2, -6, -100\).

Imagine negative numbers as a way of moving leftward on the number line. The further you go left, the smaller the number becomes.
Hence, understanding the "flow" of the number line can make it easier to grasp why \(-6\) is indeed less than \(0\).

Utilizing a Number Line

A number line is a visual representation that makes it easier for us to see how numbers relate to one another. It's like a ruler for numbers, stretching infinitely in both directions.
On a number line:

• Positive numbers are located to the right of zero.
• Negative numbers are located to the left of zero.

Each step to the left means the value is getting smaller.
Conversely, each step to the right means the value is getting larger. This makes it very intuitive to see why \(-6\) is less than \(0\): because \(-6\) falls six steps to the left of zero on the line. Using a number line is one of the simplest ways to compare numbers and understand inequalities at a glance.

Comparing Numbers

Comparing numbers is a fundamental math skill that helps determine which numbers are smaller or larger relative to each other. This skill is useful not just in math, but in daily life as well.

• Consider using the symbols \(<\) and \(>\) for comparison:
• \(<\) indicates that one number is less than another.
• \(>\) indicates that one number is greater than another.

When you compare negative numbers with positive numbers, all negative numbers are always less than zero.
This is clearly shown by their position on the number line. Therefore, in the pair \(-6\) and \(0\), it's plain to see that \(-6 < 0\). Once you get familiar with this, you'll find that comparing numbers, even on a larger scale, becomes a breeze.