Problem 5 - Algebra 1

Question

Complete the statement using \(>\) or \(<\). \(-3 \underline{?}0\)

Step-by-Step Solution

Answer

The completed statement is -3 < 0.

Step 1: Understanding Inequalities

Inequalities are relations between two values that may be different. The inequality signs '<' mean 'less than', and '>' mean 'greater than'. When we are comparing two numbers, if the number on the left is smaller than the number on the right, we use the '<' sign. If the number on the left is larger than the number on the right, we use the '>' sign.

Step 2: Understand the Position of Numbers on the Number Line

On a number line, numbers to the right are greater than numbers to the left. Zero is the midpoint on the number line - positive numbers are to its right and negative numbers to its left. As '-3' is to the left of the '0' on this number line, '-3' is less than '0'.

Step 3: Apply Inequality Sign

From Step 2, we know that '-3' is less than '0'. So, the correct inequality sign to complete the statement '-3 ? 0' is '<'. This leads us to the completed statement '-3 < 0'.

Key Concepts

Number Line

A number line is like a ruler but for numbers. It's a visual way to show how numbers are ordered. On a number line, zero is usually in the middle. Numbers increase as you move to the right and decrease as you move to the left.

• Positive numbers are found to the right of zero.
• Negative numbers are located to the left of zero.
• The farther right a number is, the larger it is.
• The farther left a number is, the smaller it is.

Using a number line helps us visually compare numbers. For example, -3 is to the left of 0, clearly showing that -3 is smaller than 0.

Greater Than

The concept of "greater than" is crucial in understanding inequalities. When we say a number is greater than another, we mean it is larger in value. We use the symbol ">" to represent this relationship.

• If we have two numbers, and the first is farther to the right on a number line, it's greater than the second number.
• For example, 5 > 2, because 5 is to the right of 2 on the number line.

In expressions, always compare each number's position to determine which is greater.

Less Than

The "less than" concept means comparing two numbers where one is smaller. We use the symbol "<" to indicate this. If a number appears to the left of another number on the number line, it is less than that number.

• For example, -3 < 0 because -3 is to the left of 0 on the number line.
• In equations like 3 < 7, 3 is less than 7.

Incorrect use of this symbol can lead to misunderstandings, so always check which number is farther left.

Negative Numbers

Negative numbers can be tricky but are important. They represent values less than zero and are found to the left of zero on the number line.

• -1, -2, -3, ..., show order from least to greatest as they move leftward.
• Negative numbers are always less than positive numbers.
• Understanding their location on the number line helps in solving inequalities.

For example, -3 is smaller than 0 because -3 is a negative number, which is always less than positive numbers.