Problem 78 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 24+3(-5+9) $$

Step-by-Step Solution

Answer

36

Step 1: Evaluate the expression inside the parentheses

First, solve the expression inside the parentheses: \(-5 + 9 = 4\).

Step 2: Multiply the result by 3

Next, multiply the result from Step 1 by 3: \(3 \times 4 = 12\).

Step 3: Add the result to 24

Finally, add the result from Step 2 to 24: \(24 + 12 = 36\).

Key Concepts

Evaluating Expressions

When you come across an algebraic expression, your first step is always to evaluate or simplify it. This means finding its numerical value. Evaluating expressions involves performing the operations given in the mathematical statement.

For example, in the exercise \( 24 + 3( -5 + 9 ) \), you need to break it down into smaller parts. This makes the task easier to manage. Understanding and performing each arithmetic operation step by step is crucial for simplified answers.

Evaluating expressions often means adhering to a specific sequence. This sequence helps ensure you get the right result.

Order of Operations

The order of operations is a rule that defines the correct sequence to evaluate a mathematical expression. This is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

For instance, consider our exercise: \( 24 + 3( -5 + 9 ) \). According to PEMDAS:
1. **Parentheses:** Start with the expression inside the parentheses: \( -5 + 9 = 4 \).
2. **Multiplication:** Next, do the multiplication: \( 3 \times 4 = 12 \).
3. **Addition:** Finally, add the result to 24: \( 24 + 12 = 36 \).

This methodical sequence ensures that every operation is performed in the proper order.

Parentheses in Expressions

Parentheses play a critical role in algebraic expressions. They indicate which operations should be performed first. Ignoring the parentheses can lead to incorrect answers.

Let's revisit our exercise: \( 24 + 3( -5 + 9 ) \). The parentheses tell us to first solve the operation inside them: \( -5 + 9 = 4 \). Only after handling the parentheses do we move on to other operations, such as multiplication and addition.

Always look for parentheses first when simplifying expressions. They guide the proper sequence of operations and ensure accurate evaluation of the expression.