Problem 18 - 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.\(-\frac{8}{7},-\frac{3}{7}\)

Step-by-Step Solution

Answer

The inequality between the numbers \(-\frac{8}{7}\) and \(-\frac{3}{7}\) is \(-\frac{8}{7} < -\frac{3}{7}\).

Step 1: Plot the numbers on the real number line

Take a number line and mark the two numbers \(-\frac{8}{7}\) and \(-\frac{3}{7}\) on it. The values on a number line progress from negative to positive from left to right.

Step 2: Identify the position of numbers

Among \(-\frac{8}{7}\) and \(-\frac{3}{7}\), \(-\frac{3}{7}\) lies to the right of \(-\frac{8}{7}\). The number on the right is greater on the real number line.

Step 3: Determine the appropriate inequality symbol

Since \(-\frac{8}{7}\) is smaller and lies to the left of \(-\frac{3}{7}\), the inequality symbol '<' is placed between them as \(-\frac{8}{7} < -\frac{3}{7}\).

Key Concepts

The Real Number Line

Understanding the real number line is crucial when learning how to plot real numbers and grasp their relationships. Imagine the real number line as a long, horizontal line that extends infinitely in both directions. This line represents all possible values that a real number can take on, from negative infinity on the left to positive infinity on the right.

In the context of the exercise given, we are dealing with negative fractions: calculate{-\frac{8}{7}} andcalculate{-\frac{3}{7}}. Plotting these numbers involves first visualizing zero as our central reference point. Numbers to the left of zero are negative, and those to the right are positive. Fractions like the ones in our exercise can be plotted based on their values in relation to whole numbers or other fractions.

For students to improve their understanding, it is recommended to practice plotting a variety of numbers, both positive and negative, including integers and fractions. Draw your own number line, place several numbers on it, and see how they compare to each other. This exercise strengthens the concept of number magnitude and their sequential nature on the line.

Inequality Symbols

Inequality symbols are the shorthand we use to describe the relative size of numbers. The two most basic inequality symbols are '<' (less than) and '>' (greater than). These symbols help us quickly express which of two numbers is larger or smaller. For example, when comparing the numbers calculate{-\frac{8}{7}} and calculate{-\frac{3}{7}}, we use '<' to indicate that calculate{-\frac{8}{7}} is less than calculate{-\frac{3}{7}}, writing it as calculate{-\frac{8}{7} < -\frac{3}{7}}.

Remember, the symbol always 'points' to the smaller number. A good tip for students is to think of the symbol as an alligator's mouth that wants to eat the bigger number, so it always opens towards it. It's also important to note that when the inequality involves negative numbers, the number closer to zero is actually larger, which is an easily confusing point for many learners.

Comparing Rational Numbers

Rational numbers are fractions that can be expressed as a ratio of two integers. When we compare rational numbers, we're looking at their size relative to each other. The task is to decide which is larger or smaller, or if they're equal. This comparison can sometimes be tricky, especially when dealing with negatives and unlike denominators.

In our example, calculate{-\frac{8}{7}} and calculate{-\frac{3}{7}} are both negative rational numbers. To compare them, it's helpful to look at their numerators since they share the same denominator. Since -8 is less than -3, and they both have a -1 factor in common (due to the negative sign), calculate{-\frac{8}{7}} is less than calculate{-\frac{3}{7}}. A useful tip for students is to practice by finding a common denominator for unlike fractions and then ordering them. Additionally, conceptualizing them on a number line, as mentioned earlier, can be very helpful.