Problem 25 - College Algebra

Question

Simplify the given expression. $$ 2 \cdot 4-9(-1) $$

Step-by-Step Solution

Answer

The simplified expression is 17.

Step 1: Resolve the Multiplication

First, handle the multiplication in the expression. Calculate \( 2 \times 4 \), which results in \( 8 \). Thus, our expression becomes \( 8 - 9(-1) \).

Step 2: Resolve the Subtraction and Multiplication of Negative Numbers

Next, deal with the subtraction and multiplication. The expression \(-9(-1)\) is multiplication, and two negatives make a positive, so \(-9(-1)\) results in \(+9\). This changes our expression to \(8 + 9 \).

Step 3: Perform the Addition

Finally, add the terms together: \(8 + 9 = 17\). Hence, the simplified expression is \(17\).

Key Concepts

Order of Operations

One crucial concept in simplifying expressions is understanding the order of operations. This universal order is often remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

When simplifying an expression, always start with any operations inside parentheses. Then, proceed with exponents, followed by multiplication and division from left to right. Finally, handle addition and subtraction, again from left to right.
Particularly, in our exercise, multiplication is handled before addition, even though addition appears first if you read the expression from left to right in the format: \(2 \cdot 4 - 9(-1)\). Observing the order of operations ensures calculations are performed correctly and consistently.

Multiplication of Integers

Let's focus on multiplying integers. Remember, multiplication of integers follows specific rules:
1. When you multiply two positive numbers or two negative numbers, the result is positive.2. When you multiply a negative number by a positive number, the result is negative.In our specific example:

• Calculate \(2 \times 4\), both integers being positive, resulting in \(8\).
• Then handle \(-9(-1)\), which involves two negative integers resulting in a positive \(+9\).

Understanding these basic rules helps simplify expressions accurately. The expression \(-9(-1)\) demonstrates how negative times negative results in positive, a fundamental concept in multiplication.

Addition of Integers

Adding integers involves combining positive and negative numbers according to their signs. Here are the key points to remember:

• Adding two positive numbers results in a positive sum.
• Adding two negative numbers results in a negative sum.
• Adding a positive and a negative number involves taking the absolute value of both, then subtracting the smaller from the larger, using the sign of the larger absolute value.

In our exercise, after resolving the multiplication, the expression becomes \(8 + 9\).
This process is straightforward as both numbers are positive, leading to their direct addition: \(8 + 9 = 17\).With these steps understood, simplifying expressions becomes predictable and methodical, ensuring the correct result each time.