Problem 82 - Contemporary Precalculus

Question

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |-3| ________ 0

Step-by-Step Solution

Answer

Question: Determine the correct inequality symbol to place between the absolute value of -3 and 0. Answer: >

Step 1: Find the absolute value of -3

The absolute value of a number is its distance from 0 on the number line. It is always non-negative. In this case, we have |-3|. Since -3 is 3 units away from 0, the absolute value of -3 is 3. So, we have: |-3| = 3

Step 2: Compare the result with 0

Now that we have found the absolute value of -3 (which is 3), we need to compare it to 0. Since 3 is greater than 0, the correct inequality symbol to use is \(>\). |-3| \(>\) 0

Key Concepts

Absolute Value

The concept of absolute value is fundamental in understanding various mathematical principles, particularly when dealing with real numbers. It's defined as the distance of a number from zero on the number line, regardless of direction. Put simply, it transforms any negative number into its positive counterpart. For example, the absolute value of \( -3 \) is \( 3 \) because if you were to measure how far \( -3 \) is from \( 0 \) on the number line, the answer would be three spaces to the left. However, with absolute value, we are really asking how far it is without considering direction, hence the answer is simply \( 3 \) units. Likewise, the absolute value of \( 3 \) is also \( 3 \), since it's three spaces to the right of zero on the number line.

It's important to remember that the absolute value output is always non-negative. Therefore, when solving absolute value equations or inequalities, it's useful to think about the original number's proximity to zero without regard to whether it's to the left or right on the number line.

Inequalities

Inequalities are mathematical expressions describing the relative size or order of two values. They're denoted by symbols like \( > \) for 'greater than', \( < \) for 'less than', \( \geq \) for 'greater than or equal to', and \( \leq \) for 'less than or equal to'. When we deal with absolute values, we often encounter inequalities. The exercise requires comparing the absolute value of a number with another number, and the understanding of inequalities is essential to solve this type of problem.

For instance, after computing the absolute value, expressing how it relates to another number with the appropriate inequality symbol is crucial. Always ensure that every step in your calculations considers whether each term is greater than, less than, or equal to the others involved. This careful analysis allows you to create a true statement about the relationship between the numbers based on their sizes.

Number Line

A number line is a visual representation of numbers on a straight line, where each point corresponds to a number. This line stretches indefinitely in both directions, with zero at the center. Numbers to the right of zero are positive, and numbers to the left are negative. Number lines are especially helpful when working with absolute value and inequalities because they provide a straightforward way to visualize the distance of numbers from zero and their relationship with each other.

When you place numbers on the number line, you can quickly gauge which numbers are larger or smaller and by how much. This makes comparing numbers and understanding inequalities much more intuitive. In the context of the given exercise, by locating \( -3 \) on the number line and noting that it lies three units to the left of zero, it's clear that its absolute value, which is the distance to zero, will be \( 3 \), reflected in the inequality \( |-3| > 0 \). Always visualize these translations on the number line to improve your grasp of absolute values and inequalities.