Problem 22 - Fundamentals of Mathematics

Question

For the pairs of real numbers in the following 5 problems, write the appropriate symbol \((<,>,=)\) in place of the \(\square\) $$ -5 \square 0 $$

Step-by-Step Solution

Answer

-5 < 0

Step 1: Understand the symbols

The task requires us to compare two numbers and use one of the three symbols: less than ( < ), greater than ( > ), or equal to ( = ).

Step 2: Identify the Numbers

In this exercise, we have two numbers: −5 and 0 .

Step 3: Compare the Numbers

Determine the relationship between −5 and 0 . Since 0 is greater than any negative number, −5 is less than 0 .

Step 4: Choose the Correct Symbol

Since −5 is less than 0 , we use the symbol < to replace the square. Therefore, −5 < 0 is the correct expression.

Step 5: Finalize the answer

The expression correctly completed is −5 < 0 .

Key Concepts

The Less Than Symbol

The "less than symbol" is represented as \(<\). It is used in mathematics to show that one number is smaller than another number. When comparing two numbers, if the first number is smaller than the second, you use the "less than symbol" between them. For example, in the situation of -5 and 0:

• -5 is less than 0, thus: \(-5 < 0\)
• This is read as "negative five is less than zero"

To make comparisons:

• Consider the number line, where numbers increase from left to right.
• -5 is at a position to the left of 0, indicating it is smaller.

Whenever you see a pair of numbers,

• Compare which is at a higher position on the number line.
• If the first number is to the left, it is less than the second.

The Greater Than Symbol

The "greater than symbol" is signified by \(>\). It is used when one number is larger than another. For example, if you have numbers like 5 and 1, the use of the greater than symbol would be:

• 5 is greater than 1, written as \(5 > 1\)
• This is read as "five is greater than one"

To properly apply the "greater than symbol":

• Visualize a number line.
• Remember, numbers that are positioned to the right are greater.
• This means 5 is to the right of 1, hence 5 is greater.

When comparing two numbers, always

• Identify which number sits further to the right.
• If the first number is more right, use \(>\).

The Equal To Symbol

The "equal to symbol" is noted by \(=\). It stands for two values being the same or identical. This symbol is used when comparing two numbers that have the same value or equivalence. An example of usage would be:

• If you compare 3 and 3, you have \(3 = 3\)
• This is read as "three is equal to three"

Using the "equal to symbol":

• Make sure both numbers have exactly the same value.
• It doesn't matter where they are on the number line if they appear exactly alike.

This concept:

• Is often seen when dealing with equations and identities where balancing both sides is necessary.
• The equality indicates absolute parity between the two entities in question.