Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Plot the two real numbers on the real number line and place the appropriate inequality symbol \((<\) or \(>)\) between them.\(-4,-8\)

Step-by-Step Solution

Answer

-4 > -8

Step 1: Plot the Real Numbers

Draw a straight horizontal line and mark it as the real number line. Mark zero at the center. To the right of zero, mark positive numbers, and to the left, mark negative numbers. Now mark -4 and -8 on this line. Remember that -8 is to the left of -4 on the number line.

Step 2: Determine the Inequality

Inequalities describe the relative sizes of different numbers. In this case, since -4 is to the right of -8 on the number line, we say that -4 is greater than -8. Therefore, we use the '>' symbol for the inequality between -4 and -8.

Key Concepts

Understanding the Real Number Line

The real number line is a fundamental concept in mathematics that helps visualize and understand both positive and negative numbers. Imagine a horizontal line stretching infinitely in both directions. This line represents all possible real numbers, with zero placed at the center.

• To the right of zero, we position positive numbers, increasing from 1, 2, 3, and so on.
• To the left of zero, we place negative numbers, decreasing such as -1, -2, -3, and beyond.
• Every point on the line corresponds to a real number, including fractions and irrational numbers.

Using this line, we can easily compare and visualize numbers. Larger numbers are to the right of smaller numbers, and similarly, more negative numbers are to the left of less negative numbers. So, on the real number line, -4 is to the right of -8.

Plotting Numbers on the Line

Plotting numbers on the real number line involves identifying their proper placement relative to zero and each other. To proceed, draw the number line and mark zero at the center.

• Locate -4 and -8 on this line by moving to the left from the zero point because both numbers are negative.
• -8 is further left on the line than -4, as -8 is less than -4 numerically.
• The further away a negative number is from zero, the smaller its value appears on the number line.

This visual placement aids in quickly determining the relationship between numbers and enables easier comprehension of concepts like inequalities.

Comparing Numbers with Inequalities

Inequalities are mathematical expressions that show the relationship between two numbers, indicating whether one is larger, smaller, or equal to the other. In our exercise, the task involves comparing -4 and -8 on the real number line.

• Inequality symbols include:
  • '<' (less than)
  • '>' (greater than)
• Since -4 is to the right of -8 on the number line, this means -4 has a greater value than -8.
• This leads us to conclude
  • -4 > -8,
  showing that -4 is indeed greater than -8.

Understanding this comparison lays the groundwork for solving various mathematical problems that involve inequalities, enhancing number sense and analytical skills.