Problem 49 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. 4 and \(-3\)

Step-by-Step Solution

Answer

The two inequalities that compare -3 and +4 are '-3 < +4' and '+4 > -3'.

Step 1: Set Up the Number Line

Create a number line. Number lines usually start from '-10' and end at '+10', this ensures that the values in question, -3 and +4, will be covered. Mark each point on the number line.

Step 2: Graph the Numbers on the Number Line

Place '-3' three steps to the left of '0' and '+4' four steps to the right of '0' on the number line.

Step 3: Write Inequalities

According to the positions of '-3' and '+4' on the number line, '-3' is less than '+4' hence the inequality '-3 < +4'. And if compared the other way, '+4' is greater than '-3' resulting in the inequality '+4 > -3'.

Key Concepts

Understanding Inequalities

Inequalities are like tiny puzzles that show how numbers relate to each other. They use symbols to express whether a number is smaller, larger, or not equal to another.

• The symbol < (less than) tells us that the number on the left is smaller than the number on the right.
• The symbol > (greater than) indicates the number on the left is larger than the one on the right.

In the example with the numbers 4 and -3, we can write two inequalities:

• -3 < 4: This tells us that -3 is smaller than 4.
• 4 > -3: This shows that 4 is larger than -3.

Understanding these symbols helps us compare values easily and accurately on a number line.

Graphing Numbers on a Number Line

When visualizing numbers, a number line serves as a helpful tool. It’s like a horizontal ruler where numbers get mapped according to their value. Here’s how to graph numbers like 4 and -3:

• Start by drawing a straight horizontal line, which will serve as the number line. Mark evenly spaced points representing whole numbers, typically going from -10 to 10 for clarity.
• Identify the number zero, which is the central point of balance on the line. This is your starting reference.
• Place the number -3 three steps to the left of the zero and the number 4 four steps to the right of the zero.

This representation helps us see where numbers are located in relation to each other, making it easier to understand and compare them.

Comparing Numbers Using a Number Line

A number line is a fantastic visual tool for comparing two or more numbers. Here's why it’s so effective:

• The position of numbers on the line shows their relationship. Numbers on the left are always smaller than those on the right.
• This means if you compare two numbers, the one located further to the right is larger.

In our example, -3 is placed left of 4, visually confirming that -3 is smaller. As you move right on the number line, values increase. Thus, comparing locations provides a clear understanding of which number is greater or smaller. This concept helps in grasping the logic behind inequalities such as -3 < 4 and 4 > -3 efficiently.