Problem 54 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$(-11+5)+(-3+2)$$

Step-by-Step Solution

Answer

The simplified result is -7.

Step 1: Identify the expression

The expression given to simplify is \((-11+5)+(-3+2)\). We need to simplify this using the order of operations.

Step 2: Simplify inside the parentheses (inner expression)

First, simplify the expression inside the parentheses. The first set of parentheses is \(-11 + 5\), which you calculate as follows: \(-11 + 5 = -6\). This represents the total for the first set of numbers.

Step 3: Simplify inside the parentheses (outer expression)

Now simplify the expression in the second set of parentheses: \(-3 + 2\). Calculate this as follows:\(-3 + 2 = -1\). This is the resulting value for the second set.

Step 4: Add the results

Now, add the results from the calculations obtained in Steps 2 and 3. You have:\(-6 + (-1)\). This calculates to:\(-6 - 1 = -7\).

Key Concepts

Prealgebra

Prealgebra serves as the foundation for all higher-level math courses, introducing students to basic yet crucial mathematical concepts. It involves understanding numbers, basic operations like addition and subtraction, and the use of variables and expressions. The goal of prealgebra is to prepare students for algebra by instilling a strong grasp of mathematical reasoning and problem-solving skills.

Key topics include:

• Understanding integers and their properties. For instance, recognizing that negative numbers exist and how to operate with them.
• Learning about basic arithmetic operations and their applications. This includes tasks like adding two negative numbers or understanding how subtraction is related to addition.
• Introducing students to expressions, which are mathematical phrases that combine numbers and operation symbols, like additions and subtractions, to convey a value.

Mastering prealgebra is essential as it eases the transition into more advanced mathematics, ensuring that students have the necessary tools to tackle algebraic equations confidently.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that helps to make complex equations manageable and understandable. By simplifying, we break down a math expression into its simplest form, often by combining like terms or executing operations.

When working with expressions:

• Always follow the rules of arithmetic, which include operations such as addition, subtraction, multiplication, and division.
• Look for and combine like terms. These are terms that have the same variables raised to the same power, allowing them to be added or subtracted from each other.
• When encountering each term, respect the sign that precedes it. A negative sign can change how terms combine with others in the expression.

Simplifying expressions properly is necessary for solving equations, which is often the ultimate goal in more advanced math problems. It reduces the complexity of a mathematical expression, making it easier to identify patterns or incorrect calculations, thus leading to accurate solutions.

Parentheses in Mathematics

The use of parentheses in mathematics is crucial for organizing calculations and directing the order of operations, ensuring accuracy in solving expressions. Parentheses tell you which operations to perform first within a given expression, a guidepost for correctly simplifying and solving it.

With parentheses:

• The operation inside them is always addressed first, following the Order of Operations (PEMDAS/BODMAS).
• Using parentheses can help clarify expressions by grouping numbers and operations, or when multiple operations are involved.
• They can affect the outcome of calculations by altering the natural precedence of operations.

Understanding the role of parentheses is paramount not just for procedural accuracy, but for making sure that your calculations reflect the intended mathematical meaning of an expression. This ensures clarity and correctness in problem-solving scenarios, particularly when paired with operations like multiplication and division, which might otherwise lead to misinterpretations if performed in the wrong sequence.