Problem 86 - Algebra A Combined Function

Question

Evaluate each expression. \(-8(-3)-4(-1)\)

Step-by-Step Solution

Answer

The expression evaluates to 20.

Step 1: Simplify the first multiplication

First, evaluate the expression \(-8 \times (-3)\). A negative multiplied by a negative results in a positive. This gives \(24\).

Step 2: Simplify the second multiplication

Next, evaluate the expression \(-4 \times (-1)\). Again, a negative multiplied by a negative results in a positive. This gives \(4\).

Step 3: Subtract the results

Now, subtract the second result from the first: \(24 - 4\). This subtraction gives \(20\).

Key Concepts

Multiplication of Integers

Multiplication of integers is one of the foundational operations in algebra. It involves combining integers, which are whole numbers, in a particular arithmetic operation. When multiplying two integers, the focus is not just on the numerical values but also on their signs. Here's a simple rule to remember:

• Multiplying two positive integers yields a positive product.
• Multiplying a positive integer with a negative integer results in a negative product.
• Multiplying two negative integers gives a positive product.

In our example,

• \(-8 imes (-3) = 24\).
• \(-4 imes (-1) = 4\).

Both multiplications involve negative integers, and according to our rule, the results are positive.

Integer Operations

Integer operations encompass addition, subtraction, multiplication, and division of integers. Unlike other numbers, integers can be negative or positive, adjusting the operation's result based on their signs. It's crucial to understand how these operations work to solve algebra problems effectively.
In the given exercise, after performing multiplications, another operation, subtraction, is used. Starting with the results of the multiplication steps:

• Multiplication yields \(24 + 4\), which we found already.
• The final subtraction step is highlighted as \(24 - 4\), leading to \(20\).

Handling integer operations accurately requires a solid grasp of the rules dictating each arithmetic operation.

Negative Numbers

Negative numbers are a crucial part of the integer number set, representing values less than zero. These numbers are indicated by a minus sign (\(-\)) before them. Negative numbers follow specific rules, especially when involved in arithmetic operations like multiplication.
When two negative numbers are multiplied, the negatives cancel out, resulting in a positive product. This might seem counterintuitive initially, but it helps maintain consistency across mathematical operations.
Understanding how negative numbers interact with each other can significantly enhance your ability to solve algebraic problems. In our exercise, we multiply negatives and confirm their product as a positive number, such as with:

• \(-8\) and \(-3\) multiply to become \(24\).
• \(-4\) and \(-1\) multiply to become \(4\).

These rules are essential to remember when you're dealing with any integer-based mathematical expressions.