Problem 34 - Fundamentals of Mathematics

Question

Determine each of the values. $$ |-2|-|-9| $$

Step-by-Step Solution

Answer

The value is -7.

Step 1: Understand Absolute Values

The absolute value of a number represents its distance from 0 on the number line, regardless of direction. Therefore, both positive and negative numbers evaluate to positive values under absolute value bars. This means \(|-2| = 2\) and \(|-9| = 9\).

Step 2: Substitute Absolute Values

Substitute the evaluated absolute values back into the expression. Replace \(|-2|\) with 2 and \(|-9|\) with 9. This changes the original expression \[ |-2| - |-9| \] into \[ 2 - 9 \].

Step 3: Evaluate the Expression

Perform the subtraction to evaluate the expression. Subtract 9 from 2, resulting in \(-7\). Thus, the value of the original expression \( |-2| - |-9| \) is \(-7\).

Key Concepts

Understanding the Number Line

The number line is an essential concept that helps us visualize numbers and their absolute values. Imagine a straight horizontal line with zero in the center. Positive numbers extend to the right, and negative numbers stretch out to the left. The further to the right a number is, the larger its value. The further to the left, the smaller its value. This visual representation helps us see how numbers relate to one another and aids in understanding absolute value.
Absolute value tells us how far a number is from zero on the number line, no matter if it's to the left or right. For instance,

• The number -2 is two units away from zero. Its absolute value is 2.
• Similarly, -9 is nine units away from zero, making its absolute value 9.

Always remember, absolute values are non-negative because they represent a distance, which never has a direction or sign.

Expression Evaluation with Absolute Values

Once we understand absolute values and the number line, evaluating expressions becomes easier. The key is to replace each number with its absolute value first. This process is called 'substitution.'
In our expression \[ |-2|-|-9| \]we first determine the absolute values:

• \( |-2| = 2 \)
• \( |-9| = 9 \)

With these values, we've transformed our original expression into another one: \[ 2 - 9 \]. Substitution helps simplify expressions, especially when dealing with absolute values, by converting everything into positive terms. This simplification makes calculations straightforward.

Simplifying Through Subtraction

Subtraction is one of the basic arithmetic operations you'll encounter when evaluating expressions. Think of subtraction as taking away a number from another. When you see \(2 - 9\), this is what you're asked to do: take 9 away from 2.
On the number line, moving left indicates subtraction. Starting from 2, you move 9 units to the left. This movement lands you at -7. Thus, \(2 - 9 = -7\).
Remember, subtraction can result in negative numbers, especially when a larger number is subtracted from a smaller one. Understanding this operation helps in evaluating expressions and solving math problems accurately.