Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. \(5+(6+4)-11\)

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Parentheses First

In the expression \(5 + (6 + 4) - 11\), start by simplifying the expression inside the parentheses. Here, it's \(6 + 4\), which equals \(10\). So, the expression becomes \(5 + 10 - 11\).

Step 2: Addition

After simplifying the parentheses, perform the addition. Add \(5\) and \(10\) from \(5 + 10\), which results in \(15\). Now the expression is \(15 - 11\).

Step 3: Subtraction

Finally, perform the subtraction. Subtract \(11\) from \(15\), which gives \(4\). Hence, the simplified expression is \(4\).

Key Concepts

Understanding the Order of Operations

When simplifying expressions in mathematics, following the correct order of operations is crucial. This ensures that everyone solves expressions consistently and correctly. In most cases, we follow an established sequence often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the provided problem, not every step in PEMDAS is relevant, but understanding this sequence helps prevent mistakes, especially in more complex expressions. Following the order step by step is essential. It helps break down the problem into manageable pieces and ensures accuracy in your solution.

How Parentheses Influence Calculations

Parentheses play a critical role in mathematics because they determine which parts of an equation you tackle first. When you see an expression like the one in the exercise, focus on the operations within the parentheses before anything else.In the expression \(5 + (6+4) - 11\), the sum within the parentheses, \(6 + 4\), is calculated first, giving us \(10\). Only after solving the innermost part of the expression do you move forward. This concept of handling the most nested operations first helps in disambiguating expressions into a straightforward sequence of operations. Always be on the lookout for parentheses as a signal to change the order of how you usually process mathematical operations. They help anchor complex calculations logically, ensuring that any expression's true value is revealed.

Steps for Addition and Subtraction

Once parentheses are resolved, the focus shifts to basic arithmetic operations: addition and subtraction. They follow immediately after any multiplication or division, yet within their own step, addition and subtraction are performed from left to right.Let's see how it applies to our expression. After replacing \(6 + 4\) with \(10\), we move to \(5 + 10 - 11\). We first perform the addition, resulting in \(15\). Then, solve the remaining part of the expression by subtracting \(11\) from \(15\), resulting in \(4\).Addition and subtraction should be approached in the order they appear, not as separate isolated tasks. This ensures that even if there are multiple operations in sequence, they are handled correctly, allowing for accurate solutions every time.