Problem 66 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \((8-9)(4-12)\)

Step-by-Step Solution

Answer

8

Step 1 - Simplify Each Parenthesis

Begin by simplifying the expressions inside each parenthesis. Compute the values for \(8 - 9\) and \(4 - 12\).

Step 2 - Calculate the Values

For \(8 - 9\), subtract 9 from 8: \(8 - 9 = -1\). For \(4 - 12\), subtract 12 from 4: \(4 - 12 = -8\).

Step 3 - Multiply the Results

Now multiply the results from Step 2: \(-1 \times -8\). The product of two negative numbers is positive, so \(-1 \times -8 = 8\).

Key Concepts

Simplifying Expressions

Before you can solve any algebraic expression, it's crucial to simplify it first. Simplifying means making the expression as easy to work with as possible.
For example, let's look at \( (8-9)(4-12) \). We start by simplifying the expressions inside each parenthesis.
The first parenthesis: \(8 - 9 = -1\)
The second parenthesis: \(4 - 12 = -8\)
So, the simplified expression is \((-1)(-8)\).
Once each part is simplified, you can then carry out further operations like addition, subtraction, multiplication, or even more complex algebraic operations. This step-by-step simplification is critical for avoiding mistakes later on.

Negative Numbers

Understanding how to work with negative numbers is an essential part of basic algebra.
Negative numbers are those less than zero and are often used to represent loss or decrease. For example, when you subtract a larger number from a smaller one: \(8 - 9 = -1\)
Here are some common points to remember about negative numbers:

• Negative numbers have a minus sign (-) before them.
• When subtracting larger numbers from smaller ones, the result is negative.

So, when handling expressions with negative numbers, always pay close attention to the signs to avoid mistakes.

Multiplication of Negatives

The multiplication of negative numbers can sometimes be tricky, but with practice, it becomes simpler.
A fundamental rule is:

• The product of two negative numbers is a positive number.
• The product of one positive and one negative number is a negative number.

Let's take \(-1 \times -8\) as an example. According to the rule, the product of these two negatives is \(+8\).
Conversely, if we had \(-1 \times 8\), the result would be \(-8\).
Remembering these rules can simplify many algebra problems and help you get the right answer more consistently.