Problem 18 - College Algebra

Question

Simplify the given expression. $$ 9-(3+11) \cdot 2 $$

Step-by-Step Solution

Answer

The simplified expression is -19.

Step 1: Evaluate the Expression Inside the Parentheses

First, we need to simplify the expression inside the parentheses. The expression inside is \(3 + 11\). Calculate this sum: \(3 + 11 = 14\).

Step 2: Perform the Multiplication

Now that we have simplified the parentheses to 14, we perform the multiplication. The expression now looks like: \(9 - 14 \cdot 2\). Calculate \(14 \cdot 2 = 28\).

Step 3: Perform the Subtraction

Finally, subtract the result of the multiplication from 9: \(9 - 28\). Calculate the subtraction: \(9 - 28 = -19\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial for solving algebraic expressions correctly. In mathematics, we follow a specific sequence to ensure the computation is done accurately. The acronym PEMDAS can help remember this sequence:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the exercise given, the expression is simplified by following these steps. We first deal with the parentheses, simplifying any calculations inside them. Then, any multiplication or division should be handled before moving on to addition or subtraction, ensuring the accurate evaluation of the expression.

Simplification

Simplification is the process of making an expression easier to work with. Here, simplifying begins with addressing the expression within the parentheses. The given example, \(9-(3+11) \cdot 2\), illustrates this well.
We start by evaluating the expression inside the parentheses, \(3 + 11\). This yields \(14\). Moving to the next step, the multiplication of \(14\) by \(2\) simplifies the expression further. This forms a simpler subtraction operation: \(9 - 28\).
This method of step-by-step simplification allows us to manage complex calculations efficiently, breaking them into smaller, more manageable parts.

Negative Numbers

Negative numbers are less intuitive than positive ones, causing confusion for many students. They can represent various concepts like debt, loss, or temperatures below zero.
In our exercise, when we compute \(9 - 28\), we are asked to subtract a larger number from a smaller one, resulting in \(-19\). The result is negative because \(28\) is greater than \(9\). To visualize this, consider starting at \(9\) on a number line and moving 28 steps to the left; you end up at \(-19\).
Understanding negative numbers is essential for accurate calculations, especially in algebra, where they frequently appear. Remember, subtraction involving larger numbers can often lead to negative results, and recognizing this pattern helps in problem-solving.