Problem 140 - Algebra A Combined Function

Question

Write each as an algebraic expression. Then simplify the expression. Twice the sum of -3 and -4

Step-by-Step Solution

Answer

The simplified expression is -14.

Step 1: Understand the Problem

The problem asks to write an algebraic expression for 'twice the sum of -3 and -4'. We need to express this mathematical sentence using symbols and then simplify it.

Step 2: Translate Words to an Expression

The phrase 'twice the sum of -3 and -4' can be translated to an algebraic expression. 'The sum of -3 and -4' is written as \(-3 + (-4)\). 'Twice' indicates that this sum should be multiplied by 2. Thus, the complete expression is \(2 \times (-3 + (-4))\).

Step 3: Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses. Calculate \(-3 + (-4)\). This sum equals \(-7\).

Step 4: Multiply to Simplify

Next, multiply by 2 to simplify the expression. Compute \(2 \times (-7)\), which is equal to \(-14\).

Key Concepts

Simplifying Expressions

When we simplify expressions, the goal is to make them as concise and clear as possible. In algebra, this often means combining like terms and performing basic arithmetic. For instance, simplifying \(2 \times (-3 + (-4))\) involves two main steps.
First, tackle the operations within the parentheses, which requires knowing how to handle addition, especially with negative numbers. Solve \(-3 + (-4) = -7\).
Then, finish by multiplying \(2 \times (-7)\) to get a final simplified result of \(-14\).
A simplified expression is easier to comprehend and use in further mathematical calculations.

Translating Phrases to Expressions

Translating verbal phrases into algebraic expressions is a foundational skill in algebra. It is like turning words into numbers and symbols, a crucial step for solving real-world problems.
Consider the phrase "twice the sum of -3 and -4." We break this down:

• 'The sum of -3 and -4' translates to \(-3 + (-4)\).
• The word 'twice' means to multiply the sum by 2, presenting us with the expression \(2 \times (-3 + (-4))\).

Mastery in translating phrases helps in setting up equations correctly, making the rest of the solving process smoother.

Negative Numbers

Negative numbers often appear in everyday tasks like calculating debts or temperatures below zero. In algebra, understanding them is crucial for accurately simplifying and solving expressions.
When dealing with negative numbers in expressions, especially in addition and multiplication, remember these rules:

• Adding negatives: Adding two negative numbers results in a more negative number, such as \(-3 + (-4) = -7\).
• Multiplying by a negative number flips the sign. For example, \(2 \times -7\) becomes \(-14\).

Getting comfortable with negative numbers is key to success in algebra.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division are the backbone of algebra. Knowing the order and rules of these operations can make solving algebraic expressions straightforward.
In the expression \(2 \times (-3 + (-4))\), multiple operations are involved:

• Order of Operations: Parentheses first, then multiplication.
• Addition: \(-3 + (-4)\) simplifies to \(-7\).
• Multiplication: Multiply the result, \(-7\), by 2 to obtain \(-14\).

Following the order and correctly performing each operation leads to accurate solutions.