Problem 69 - Elementary Algebra

Question

\(5-|-9|\)

Step-by-Step Solution

Answer

-4

Step 1: Understand the Absolute Value

Identify the absolute value expression in the problem. Here, it is \(|-9|\). The absolute value of a number is the distance between that number and zero on a number line, regardless of direction. Thus, \(|-9| = 9\).

Step 2: Substitute the Absolute Value

Replace the absolute value expression \(|-9|\) with its evaluated value. The expression then becomes \(5 - 9\).

Step 3: Simplify the Expression

Perform the subtraction operation. \(5 - 9\) equals to \(-4\).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on a number line. Unlike normal values, it does not take direction into account. For example, \(|-9|\) is the absolute value of \-9\, which equals 9 because it’s 9 units away from zero. Similarly, \(|5|\) is just 5. Absolute value helps simplify expressions involving both positive and negative numbers. This way, we treat all numbers as non-negative distances for easier calculations.

Key Points:

• Absolute value always returns a non-negative number.
• Represented by vertical bars around the number, like \(|x|\).
• Useful in calculating differences and distances.

Subtraction

Subtraction means taking away one number from another. It’s like moving left on the number line. In this exercise, after evaluating the absolute value, we substitute it back into the expression to get \(5 - 9\). Key to remember:

• Subtracting a larger number from a smaller one results in a negative value.
• Order of subtraction matters: \(5 - 9 ≠ 9 - 5\).
• Always perform operations from left to right, following BODMAS/BIDMAS rules.

Important points here:

• Recognize whether subtraction will lead to positive or negative results.
• Consistently apply subtraction rules to avoid errors.

Number Line

A number line is a visual tool that helps understand numbers and their relationships. It extends infinitely in both positive and negative directions. When we talk about absolute value, we refer to the distance on this line.

Let’s break it down with \(5 -|-9|\). First, find the absolute value: \(|-9| = 9\). Now, use the number line to see the subtraction plan: start at 5 and move left 9 places to reach -4.

• Number lines help visualize arithmetic operations and their results.
• They make understanding positive and negative values easier.
• A clear mental image of moving points on the number line aids in grasping concepts better.