Problem 58 - Elementary Algebra

Question

Simplify. $$ 9-4|-3| $$

Step-by-Step Solution

Answer

The simplified expression is -3.

Step 1: Evaluate the Absolute Value

First, we identify the expression inside the absolute value bars: \(-3\). Since absolute values turn negative numbers into positive numbers, \(|-3| = 3\).

Step 2: Substitute and Simplify

We substitute the evaluated absolute value back into the expression, replacing \(|-3|\) with \(3\): \(9 - 4 \times 3\).

Step 3: Perform Multiplication

Next, calculate \(4 \times 3\): \(4 \times 3 = 12\).

Step 4: Perform Subtraction

Finally, subtract the result from the multiplication from 9: \(9 - 12 = -3\).

Key Concepts

Understanding Absolute Value

The absolute value of a number is essentially the number's distance from zero on the number line, without considering its direction. Think of it as only caring about how far you are from zero—not whether you're on the negative or positive side. For any positive number, the absolute value is the number itself. For instance,

• The absolute value of 5 is 5, and
• The absolute value of -5 is also 5.

This is because -5 is 5 units away from zero on the number line; we just ignore the negative sign because distance cannot be negative. In mathematical expressions, absolute value is shown using vertical bars, for example, \[|-3| = 3\] This simple operation helps deal with a lot of real-world situations where only magnitudes are important, like in measuring distances or determining time durations.

The Art of Simplification

Simplification in mathematics involves reducing an expression to its simplest form. Simplifying expressions makes them easier to understand and work with. The process of simplification follows a sequence of operations according to rules of arithmetic and algebra, which we call the "Order of Operations." This includes handling absolute values, multiplication, division, addition, and subtraction, in the right order. For the expression in our exercise, by replacing \[|-3|\] with its absolute value, 3, we get a clearer view of the expression. It allows you to easily carry out further operations like multiplication and subtraction, leading to the final simplified result:\[9 - (4 \times 3) = 9 - 12 = -3\]The goal of simplification is usually to make calculations faster and results easier to interpret.

Basics of Arithmetic Operations

Arithmetic operations form the foundation of mathematics and include addition, subtraction, multiplication, and division. These operations are not only the building blocks of algebraic expressions but are also essential in everyday calculations. In the exercise, we needed to apply two basic operations:

• Multiplication: We multiplied 4 by 3 to get 12.
• Subtraction: We then subtracted the result, 12, from 9 to get -3.

It’s important to note the order in which operations are performed, which follows the "PEMDAS/BODMAS" rule: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding how and when to apply these operations is crucial in solving more complex expressions efficiently.