Problem 105 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ (8-11)(9-12) $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1 - Simplify inside the parentheses

First, simplify the expressions inside the parentheses. Calculate the value of each subtraction: \(8 - 11 = -3\) and \(9 - 12 = -3\).

Step 2 - Multiply the simplified values

Next, multiply the results from the simplifications: \((-3) \times (-3)\).

Step 3 - Simplify the multiplication

Finally, simplify the multiplication of the values: \((-3) \times (-3) = 9\).

Key Concepts

headline of the respective core concept

When simplifying algebraic expressions, it is crucial to understand the basic operations involved. These operations include addition, subtraction, multiplication, and division. We often deal with expressions that require us to apply these operations in a specific sequence, known as the order of operations. Always remember: PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This will help you keep the proper structure as you simplify any expression.
Let's focus on three key concepts to solve the exercise given above: algebraic operations, parentheses in expressions, and multiplication of integers.

headline of the respective core concept

Using parentheses in expressions helps us determine which operations to perform first. According to the order of operations (PEMDAS), we always start with calculations inside the parentheses.
In our exercise, we have two parentheses to deal with: \((8-11)\) and \((9-12)\).
We simplify these by performing the subtractions inside each one, leading to: \(8 - 11 = -3\) and \(9 - 12 = -3\).
Thus, our expression now becomes: \((-3)(-3)\).
Notice how handling the parentheses first simplified our problem significantly.

headline of the respective core concept

Next, let's explore the multiplication of integers. Multiplying two negative integers may seem tricky at first, but there's a simple rule: The product of two negative integers is positive. In our exercise, we need to multiply \(-3 \times -3\).
When two negative values are multiplied, the negatives cancel each other out, resulting in a positive product. Hence, \(-3 \times -3 = 9\).
This final step provides the simplified value of our original expression. Understanding the multiplication of integers and how negatives interact can make problems like this easier to handle.