Problem 52 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$(-3+1)+(-9+4)$$

Step-by-Step Solution

Answer

The simplified expression is -7.

Step 1: Identify Operations Inside the Parentheses

First, we need to address the expressions inside the parentheses. There are two separate expressions: 1. \[ (-3 + 1) \] 2. \[ (-9 + 4) \] These operations should be performed first according to the order of operations (parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)).

Step 2: Simplify Expression in the First Parenthesis

For the expression inside the first parenthesis: \[ (-3 + 1) \]Simplifying, we get:\[ -3 + 1 = -2 \]

Step 3: Simplify Expression in the Second Parenthesis

For the expression inside the second parenthesis: \[ (-9 + 4) \]Simplifying, we get:\[ -9 + 4 = -5 \]

Step 4: Combine the Results from Each Parenthesis

Now, we add the results of the simplified expressions from each parenthesis: \[ (-2) + (-5) \]Simplifying this, we get:\[ -2 - 5 = -7 \]

Key Concepts

Simplifying Expressions

When simplifying expressions, especially those involving various operations, it is key to proceed step-by-step following a set system to ensure accuracy. This often involves the order of operations, which provides a clear map of how to tackle the components of an expression.

• Focus on simplification of smaller parts, like separate terms or components in parentheses.
• Address operations in the sequence prescribed by the order of operations, which is often abbreviated as PEMDAS - Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right).

In the given exercise, the initial step of simplification involves solving each bit within the parentheses. By breaking down complex expressions into smaller, manageable parts, we can streamline the entire process, reducing the potential for errors and misunderstandings.

Parentheses

Parentheses are like traffic lights for mathematic operations; they tell us where to stop first. Whenever you see parentheses, focus on them before moving on to the rest of the expression. Solving expressions inside parentheses first is crucial because it dictates the outcomes of the broader expression.Let's look at using parentheses in our exercise:

• The expression begins with \((-3+1)\) and \((-9+4)\), both enclosed in parentheses.
• Each needs to be simplified separately before they can be combined.

By first addressing what's inside the parentheses, you ensure that every part of the expression is evaluated correctly, maintaining mathematical accuracy.

Addition of Integers

Handling the addition of integers, particularly with positive and negative numbers, can sometimes be a source of confusion. However, it is often straightforward when you follow a few basic principles:

• Consider the signs: Adding two negative numbers like \(-2 + (-5)\) results in a more negative number, hence \(-2 - 5 = -7\).
• When numbers have different signs, like \(-3 + 1\), find the difference between their absolute values and take the sign of the number with the larger absolute value. This means \(-3 + 1\) results in \(-2\).

Mastering these guidelines enables you to tackle integer addition confidently, ensuring you handle even complex expressions with ease. Remember, practice makes perfect, and familiarizing yourself with how integers interact will lead to smoother and quicker solutions.