Problem 41 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$3(-2) 4+3(-2)$$

Step-by-Step Solution

Answer

The simplified expression is -30.

Step 1: Identify Operations and Grouping

Look at the expression: \( 3(-2) \cdot 4 + 3(-2) \). Notice the operations involved: multiplication and addition. There are no parentheses affecting order aside from the implicit ones in multiplication with negative numbers. According to the order of operations, we perform multiplication before addition.

Step 2: Perform the Multiplications

Calculate the multiplication operations first. \( 3(-2) \cdot 4 = -6 \cdot 4 = -24 \). Also, calculate \( 3(-2) = -6 \).

Step 3: Perform the Addition

Now substitute the calculated values back into the expression. Add the two results: \( -24 + (-6) = -24 - 6 = -30 \).

Key Concepts

Understanding Multiplication in Order of Operations

When it comes to solving mathematical expressions, multiplication plays a crucial role and often comes first in the order of operations, commonly remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication & Division (from left to right), Addition & Subtraction (from left to right). In this step-by-step process, multiplication is considered a higher priority than both addition and subtraction. Hence, it occurs earlier in the process of simplification.
Let's look at the expression from the exercise:

• Start with the multiplication: In the expression \(3(-2) \cdot 4\), handle the \(3(-2)\) first, which equals \(-6\).
• Next, multiply \(-6\) by \(4\) to get \(-24\).
• This part of the expression simplifies the former's complexity before moving onto other operations like addition.

Recognizing that multiplication sometimes involves negative numbers—as shown by the negative symbol before the 2—is also important, affecting the sign of the overall result. When you multiply two numbers with different signs, the result is negative.

Adding with Negative Numbers

Addition is a simple operation, but it can become tricky when negative numbers are involved. Adding a negative number is similar to subtracting a positive number, which can be difficult to wrap the head around if unfamiliar.
For example: In the exercise, we end up adding the results of two multiplications: \(-24\) and \(-6\). Rather than simply moving numbers around, adding these two is like combining negative values. The expression \(-24 + (-6)\) is actually the same as subtracting 6 from \(-24\):

• \(-24 + (-6) = -24 - 6\)
• The result is \(-30\), indicating a further shift into the negative number line.

Handling addition with negative numbers smoothly reinforces the need to take negative values into account adequately in expression simplification.

Subtraction & Its Relation to Addition

Subtraction is often taught as 'taking away' and is closely related to addition. In fact, subtraction can be considered as adding a negative number, which emulates a core concept taught in mathematics. Understanding this relationship helps in tackling expressions involving both operations.
For instance, simplifying the expression into subtraction form can initially appear unusual. Yet, when the step shows \(-24 + (-6)\), it's similar to reorganizing the addition as a subtraction problem:

• Think of it as \(-24 - 6\), where you are subtracting 6 from \(-24\).
• The importance lies in understanding that subtraction decreases a number's value, more so along the negative scale.

Viewing subtraction as the addition of a negative number broadens comprehension of expressions involving both operations. In essence, the result of \(-30\) remains the same, just deduced through a different analytical lens.