Problem 30 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line. Then write two inequalities that compare the numbers. \(9,-12\)

Step-by-Step Solution

Answer

On the number line, -12 is to the left of 9, so we have -12 < 9. The inequalities are -12 < 9 and 9 > -12.

Step 1: Graph the Numbers

Draw a number line. Then mark -12 and 9 on the number line. The number -12 is placed to the left of 9 indicating that it is smaller than 9.

Step 2: Write the Inequalities

The number to the left is always smaller than the number to the right on the number line. So, -12 is less than 9 which can be written as () -12 < 9. Reversely, 9 is greater than -12 which can be written as () 9 > -12.

Key Concepts

Graphing Numbers

Graphing numbers is a foundational skill in mathematics that helps visualize the size and order of numbers on a number line. To start, we need to understand what a number line is. It is a straight horizontal line with evenly spaced marks or intervals that represent numbers. By convention, numbers increase from left to right.

When graphing specific numbers like \( -12 \) and \( 9 \), locate them correctly on the number line. Since \( -12 \) is negative, it will be to the left of \( 0 \). Positive numbers like \( 9 \) will appear to the right. This placement shows the relationship between the numbers visually, making comparison straightforward.

• The number line helps us see that \( -12 \) is left of \( 9 \).
• By seeing their positions, we instantly know that \( -12 \) is less than \( 9 \).

Graphing numbers tells us not just about their size, but also about their order relative to each other.

Inequalities

Inequalities help in comparing numbers by indicating one quantity is less or greater than another. Symbols like \(<\) and \(>\) are used to denote these comparisons. When using a number line, understanding inequalities becomes much simpler.

Consider the placement of numbers on the number line. The fundamental rule is:

• A number positioned to the left is always less than those further to the right.
• Conversely, a number to the right is greater than those to the left.

For example, with \( -12 \) and \( 9 \) placed on the number line, \( -12 \) is to the left of \( 9 \). Thus, we say \( -12 < 9 \).

Inequalities provide a contextual understanding of numbers beyond the visible graph, enabling meaningful comparison.

Comparing Numbers

Comparing numbers allows us to determine the relative size or value of numerical quantities. This is especially useful in real-world problem-solving when we need to assess two or more amounts.

When comparing two numbers like \( 9 \) and \( -12 \), several steps clarify their relationship:

• Visualize them on a number line to see their positions in relation to each other.
• Use inequality signs: The number to the left is smaller than the number on the right.
• Write this relationship using symbols: For instance, \( 9 > -12 \), indicating \( 9 \) is greater than \( -12 \).

Through these tools, we see how large numbers compare to smaller numbers or vice versa, aiding in decision-making and precise mathematical communication.