Problem 79 - Contemporary Precalculus

Question

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |-2| ______ |-5|

Step-by-Step Solution

Answer

Answer: |-2| < |-5|

Step 1: Understand the absolute values

The absolute value of a number is the number's distance from 0 on the number line. It is always a non-negative value, which means it does not have a negative sign. In this case, we need to compare the absolute values of -2 and -5.

Step 2: Find the absolute values of -2 and -5

The absolute value of -2, denoted by |-2|, is the distance between -2 and 0 on the number line, which is 2. The absolute value of -5, denoted by |-5|, is the distance between -5 and 0 on the number line, which is 5.

Step 3: Compare the absolute values

Now that we have the absolute values of both numbers, we can compare them: 2 and 5. Since 2 is less than 5, the correct comparison symbol is "<".

Step 4: Final answer

The correct way to complete the statement is: |-2| < |-5|.

Key Concepts

Absolute Value Definition

The concept of absolute value is foundational in understanding how to compare numerals within a mathematical context. In essence, the absolute value of a number is the distance from that number to zero on a number line, irrespective of direction. Think of it as a measure of magnitude only. Hence, by definition, absolute values can never be negative, as distance itself is inherently a non-negative quantity.

For any given number 'x', the absolute value is denoted as |x|. Let's consider a number line, which is a straight, horizontal line with zero in the center, positive numbers to the right of zero, and negative numbers to the left. The distance a number lies from zero is its absolute value. For example, both 3 and -3 are three units away from zero, so they both have an absolute value of 3, represented as |3| and |-3|, both equaling 3.

Comparing Absolute Values

When comparing absolute values, we are essentially evaluating the magnitudes of different numbers without considering their sign (positive or negative). Comparisons are crucial for understanding the relationships between quantities and are performed using the symbols < (less than), > (greater than), or = (equal to).

It is much like comparing distances: if you are three steps from a door and another person is five steps away, you are closer, and in absolute value terms, 3 is less than 5. Conversely, if you had taken five steps and your friend only three, you would be farther from the door, and 5 is greater than 3 in absolute value terms. The same logic applies to negative numbers, so |-3| is less than |-5| because 3 is less than 5.

In our exercise, the task is to determine how |-2| and |-5| relate. Recall that the absolute value reflects distance without regard for direction; thus, |-2|, which equals 2, is indeed less than |-5|, which equals 5. Hence, the comparison symbol that makes the statement true is <.

Number Line Distances

A number line is an incredibly helpful visual tool to understand absolute values and compare them. It can be thought of as a real-life ruler of sorts where every number sits at a fixed distance from the 'zero' point.

Visualizing numerical distance on a number line can improve comprehension of the abstract concept of absolute value. In our exercise, illustrating -2 and -5 on a number line allows a clear view that -2 is closer to zero than -5. Therefore, its absolute value, or 'number line distance,' is smaller. This utilitarian visual approach can present a better understanding for students grappling with abstract numerical concepts and fortify their grasp on comparing absolute values.