Problem 5 - Fundamentals of Mathematics

Question

Determine each a bsolute value. $$ -|9| $$

Step-by-Step Solution

Answer

The final value is \(-9\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For the number \(9\), \(|9| = 9\).

Step 2: Apply the Negative Sign

The given expression is \(-|9|\). First, find \(|9|\), which is \(9\), and then apply the negative sign as \(-9\).

Key Concepts

Negative numbers

Negative numbers are numbers that are less than zero. They are usually found to the left of zero on the number line. When you think about temperatures, debts, or elevations below sea level, you are often dealing with negative numbers.

These numbers are quite interesting because they can be thought of as opposites of positive numbers. For example,

• -3 is the opposite of 3.
• -10 is the opposite of 10.

Whenever you see a negative sign, it means you are looking at a negative number, which is below zero when visualized on a number line. This means money owed rather than money owned or a drop in the ocean rather than an elevation.

Number line

The number line is a visual representation of numbers in a straight line. You can use it to understand positive and negative numbers better. In a number line,

• Zero is usually placed at the center.
• Positive numbers extend to the right.
• Negative numbers extend to the left.

This line helps to visualize where numbers are relative to each other and is especially helpful in understanding the concept of the absolute value, where the focus is not on which way a number is located but how far it is from zero.

Distance from zero

Distance from zero is a term often used when talking about absolute value. To put simply, the absolute value of a number is precisely its distance from the zero on a number line, without considering whether the number is positive or negative.

For example, both

• |7| = 7
• |-7| = 7

show that both 7 and -7 are seven units away from zero. This concept allows you to see numbers as a measure of space from zero, rather than just their sign. Remembering this can be very helpful, especially when solving problems that involve absolute values, because it always emphasizes distance, not direction.