Problem 49 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-4 \quad\square -6$$

Step-by-Step Solution

Answer

The correct statement is -4 > -6.

Step 1: Understanding negative numbers

When dealing with negative numbers, a smaller numerical value is actually larger. For example, -1 is larger than -2 though 1 is smaller than 2.

Step 2: Comparing the numbers

Here, -4 is being compared to -6. Since -4 is closer to zero than -6, -4 is the larger number.

Step 3: Inserting the correct sign

Considering that -4 is larger than -6, the correct sign to insert would be '>'. So, the correct statement is '-4 > -6'.

Key Concepts

Understanding Negative Numbers

Negative numbers might seem tricky at first because they operate differently from positive numbers. When comparing negative numbers, a number is less negative if its absolute value is closer to zero. Therefore:

• -2 is greater than -5.
• -3 is greater than -6.

The smaller the digit after the negative sign, the bigger the number actually is, because it means it is closer to zero. This is the opposite of how we compare positive numbers. Let's simplify it: the less negative, the bigger it is!

Basics of Number Comparison

Number comparison means figuring out which number is larger or smaller compared to another. For negative numbers, this means you must remember that the closer a number is to zero, the larger it actually is.
Compare -4 and -6. Since -4 is closer to zero than -6, -4 is considered larger in negative number terms.

• -1 > -3 because -1 is closer to zero.
• -9 < -2 because -9 is further from zero.

Always think about the distance from zero: the smaller the absolute value in negatives, the larger the actual value.

Using Mathematical Symbols for Inequalities

When comparing numbers, mathematical symbols help us to easily represent relationships between them. The symbols often used are:

• The greater than symbol \(>\) which means one number is larger than another.
• The less than symbol \(<\) which indicates that a number is smaller.

For example, when you figure out that -4 is larger than -6, you use the greater than symbol \(>\) to express this relationship: \(-4 > -6\).
In inequalities, choose the correct symbol based on your comparison. Remember:

• When you compare negative numbers, be careful not to mix up the symbols. The one closer to zero is the bigger number!

With practice, using these symbols becomes second nature!