Problem 70 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line. Then write two inequalities that compare the numbers. \begin{equation} 7,-7 \end{equation}

Step-by-Step Solution

Answer

The two inequalities comparing the numbers 7 and -7 are \(7 > -7\) and \(-7 < 7\)

Step 1: Draw the Number Line

Represent the real number system using a horizontal line. Mark a point in the middle of this line as the number zero. Then to the right and left mark points representing both positive and negative integers. Specifically, locate and mark the numbers 7 and -7 on the line.

Step 2: Plot the Numbers

From the number zero, count 7 units to the right and mark this point as the number 7. Similarly, from the number zero, count 7 units to the left and mark this point as -7.

Step 3: Formulate Inequalities

At this point, inequalities can be established following the rule that numbers to the right of a given number are greater, and numbers to the left are smaller. Hence we obtain: \(7 > -7\) and \(-7 < 7\).

Key Concepts

Number Line

A number line might seem like a simple horizontal line, but its power lies in representing the entirety of the real numbers in a visually clear way. Imagine a straight line extending infinitely in both directions, centered at zero. Each point on this line corresponds to a real number, with positive numbers to the right of zero and negative numbers to the left.

• Zero is always your starting reference point.
• Positive numbers increase in value as you move to the right.
• Negative numbers decrease in value as you move to the left.

Number lines are particularly useful tools for concepts in math such as comparison of numbers, finding distances (or magnitudes), and for beginners, understanding basic arithmetic. By simply marking out points on this line, you can represent numbers and see their clear relationship to one another, which is helpful when solving inequalities.

Integer Representation

Integers are the set of whole numbers and their negatives, including zero. They play a crucial role in understanding inequalities and the number line itself. When you plot integers on a number line, you begin to see their relationships unfold. Let's consider the integers 7 and -7, as mentioned in our original exercise.

• The integer 7 is positive and, when plotted, will be to the right of zero.
• The integer -7 is negative and will be to the left of zero.
< li>Both these integers are equidistant from zero, but on opposite sides.

This visualization aids in comprehending the essence of inequality, where 7 is larger because it is positioned to the right of -7. By placing these integers on the line, one can clearly appreciate how integers interact within the spectrum of numbers, allowing for straightforward application of mathematical operations and concepts.

Graphing Numbers

Graphing numbers on a number line is a visual method for understanding mathematical relationships. It is straightforward and incredibly helpful, especially when dealing with inequalities.To graph the numbers 7 and -7:

• Start at zero on your number line.
• Move 7 spaces to the right to plot the number 7.
• Move 7 spaces to the left to plot the number -7.

Once you've plotted these points, the graph itself communicates fundamental truths about these numbers:- The farther right a number is, the greater it is.- The farther left, the smaller it becomes.Using this graph, you can easily formulate inequalities. Since 7 is to the right of -7, it's plain to see that 7 is greater than -7, or in inequality terms, \(7 > -7\). Similarly, we observe that -7 is less than 7, or \(-7 < 7\). This simple yet powerful method of visual representation helps in grounding our understanding of numeric relationships and inequalities.