Problem 21 - Contemporary Precalculus

Question

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. 5 ______ -3

Step-by-Step Solution

Answer

Answer: The correct symbol to place between 5 and -3 is greater than (>). So, 5 > -3.

Step 1: Understand the number line

To compare the two numbers, imagine a number line where positive numbers are to the right of 0 and negative numbers are to the left of 0. The farther right a number is, the greater it is; conversely, the farther left a number is, the smaller it is.

Step 2: Locate the numbers on the number line

Now, locate the numbers 5 and -3 on the number line. The number 5 is five units to the right of 0, while -3 is three units to the left of 0.

Step 3: Compare the numbers

Since 5 is to the right of -3 on the number line, it is greater than -3.

Step 4: Fill in the blank

Now that you know 5 is greater than -3, fill in the blank with the appropriate symbol: 5 > -3.

Key Concepts

Number Line

A number line is a straight, horizontal line with numbers placed at intervals along the length of it. This is an essential tool in mathematics for understanding the relative positions of numbers.
Naturally, every number line features zero at its center. Numbers increase as you move to the right and decrease as you move to the left. This visual representation helps in understanding various mathematical concepts, including adding, subtracting, and comparing numbers.

To effectively use a number line:

• Locate numbers by starting from zero and counting units either left for negatives or right for positives.
• Remember that the direction tells us about the value. To the right means greater, to the left means lesser.

This simple concept can even help young learners grasp the idea of greater than and less than easily.

Comparing Numbers

Comparing numbers is a fundamental mathematical skill. It involves determining which number is greater, smaller, or if two numbers are equal. You can do this visually using a number line or by considering their values directly.

Key strategies for comparing numbers include:

• Place each number on the number line. Whichever is further to the right is the greater number.
• Use symbols :
  • \( < \) means less than.
  • \( = \) means equal to.
  • \( > \) means greater than.
• Always check the position of numbers relative to zero.

In the original exercise, comparing 5 and -3, placing them on the number line helped us see that 5 is further right, therefore 5 is greater than -3.

Positive and Negative Numbers

Positive and negative numbers make up the integers on a number line. Positive numbers are positioned to the right of zero, while negative numbers are found to the left. Zero itself is neither positive nor negative.

Understanding positive and negative numbers is crucial since:

• They help indicate direction or size; for example, a temperature of -3°C is 3 degrees below zero.
• In terms of value, any positive number is greater than any negative number.
• Addition and subtraction rules involve changing direction on the number line.

Knowing how to work with these numbers, especially when comparing them, is vital in solving problems like the exercise where you determine that 5 is greater than -3. Remember, a trip towards the right on the number line is always a trip to greater values.