Problem 51 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line.Then write two inequalities that compare the numbers. $$-6,3$$

Step-by-Step Solution

Answer

The position of -6 and 3 on the number line, and the inequalities are \(-6 < 3\) and \(3 > -6\).

Step 1: Draw a Number Line

To start, draw a straight line. Label it in such a way that it represents a number line and length interval goes from least (-6) to greatest (3) which includes 0.

Step 2: Plot the Numbers

Next, plot -6 and 3 on the number line. The number -6 will be to the left of 0 and 3 will be on the right of 0.

Step 3: Write Inequalities

Now, observe the relationship between -6 and 3 on the number line. Note that -6 is less than 3, and 3 is greater than -6. Write these relationships as inequalities: \(-6 < 3\) and \(3 > -6\).

Key Concepts

Graphing on a Number Line

To understand the concept of a number line, think of it as a long, straight ruler that marks different numbers in order. Each point on the number line corresponds to a number. The center of the number line is usually marked with zero. Negative numbers are to the left of zero, and positive ones are to the right.

When graphing numbers such as -6 and 3:

• Start by drawing a horizontal line and marking a point labeled 0 in the middle.
• Add marks at equal intervals both to the left and right of zero.
• To the right, label 1, 2, 3, and so on, and to the left, label -1, -2, -3, continuing until you reach -6.
• Finally, place a dot above the -6 and the 3 to indicate their positions on the line.

In this manner, you visually represent numbers in relation to each other, which is very useful for comparing values and performing mathematics operations like addition and subtraction.

Understanding Inequalities

Inequalities are mathematical expressions that describe how numbers relate to each other in terms of size. Instead of stating that two numbers are equal, inequalities tell you if one is bigger, smaller, or not as large in proportion to another. Common inequality symbols are:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

When looking at our number line example with -6 and 3:

• -6 is positioned to the left of 3, indicating that -6 is less than 3.
• This relationship can be written as the inequality \(-6 < 3\).
• The inverse is also true: 3, being to the right of -6, is greater than -6, denoted as \(3 > -6\).

These simple comparisons through inequalities help in understanding and solving mathematical problems.

Comparing Numbers

When we talk about comparing numbers, we think about their positions on the number line. This concept is fundamental in arithmetic and invaluable in various applications, such as understanding statistics or analyzing data.

Let's consider our example with numbers -6 and 3:

• -6 is to the left of 3 on the number line. Hence, -6 is smaller than 3.
• Comparing the same numbers in reverse, 3 is larger than -6 because it comes later on the number line.

In mathematics, comparing numbers might involve negative and positive numbers, zero, or even decimals and fractions. Plotting on a number line does not just show the position but also helps in visualizing differences and understanding how one number is greater or less than another. This forms a bridge to dealing with more complex math concepts effortlessly in the future.