Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ 3 m-2\\{m-3[2 m-(m-5)]+4\\} $$

Step-by-Step Solution

Answer

The simplified expression is \(7m + 22\).

Step 1: Write down the problem and focus on the innermost parentheses

First, we need to tackle the innermost parentheses so that we can use the proper order of operations: $$ 3m - 2\{m - 3[2m - (m - 5)] + 4\} $$ The innermost parentheses is \((m-5)\), we will simplify the expression inside the brackets next.

Step 2: Simplify the inside of the brackets [ ]

Now, we will distribute the 3 inside the brackets to the terms within them: $$ 3m - 2\{m - 3[2m - m + 5] + 4\} $$ Combine like terms within the brackets: $$ 3m - 2\{m - 3[m + 5] + 4\} $$

Step 3: Distribute the -3 inside the brackets {}

Let's distribute the -3 inside the brackets {} to the terms within them: $$ 3m - 2\{m - 3m - 15 + 4\} $$ Combine like terms within the brackets {}: $$ 3m - 2\{-2m - 11\} $$

Step 4: Distribute the -2 outside the brackets {}

Now distribute the -2 to the terms inside the brackets {}: $$ 3m + 4m + 22 $$

Step 5: Combine like terms

Finally, we combine the like terms, the terms that are multiples of m: $$ (3m + 4m) + 22 = 7m + 22 $$ The simplified expression is: \(7m + 22\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. The order is often described using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• Parentheses: Always perform operations inside parentheses first.
• Exponents: Solve any exponential terms next.
• Multiplication and Division: Proceed with these operations in the sequence they appear, moving left to right.
• Addition and Subtraction: Finally, handle these similarly, from left to right.

In the expression \(3m - 2\{m - 3[2m - (m - 5)] + 4\}\), following the order of operations allows us to effectively manage complexity and reach the simplified form \(7m + 22\). Each step adheres to this hierarchy ensuring that no terms are miscalculated.

Distributive Property

The distributive property is used to multiply each term inside a bracket by a number outside the bracket. It's a method that ensures every part of the expression is accounted for correctly. Formally, it is expressed as \(a(b + c) = ab + ac\).
Here’s how it works in the expression:

• First, distribute 3 within the brackets: \(3[2m - (m - 5)]\).
• Then, distribute -3 within the same set of brackets: \{-3[m + 5]\}.
• Finally, distribute -2 outside of the whole bracketed expression: \(-2\{-2m - 11\}\).

By following these steps, the expression is broken down into simpler parts, allowing us to easily simplify it to \(3m + 4m + 22\). This step-by-step approach ensures clarity and accuracy.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that makes expressions easier to work with. Like terms have the same variable part. For example, terms like \(3m\) and \(4m\) can be combined because they both contain the variable \(m\).

• Check for coefficients that have the same variable.
• Add or subtract these coefficients.

In the expression, we have \(3m + 4m\) after applying the distributive property. Combining these gives us \(7m\). By ensuring all similar terms are combined, you simplify the expression without changing its value, which helps in maintaining clarity throughout the problem-solving process.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. They can be as simple as \(x + 2\) or more complicated like the one given in our example.

• Expressions often consist of terms separated by plus or minus signs.
• Each term includes numbers, variables, or both, interacting through multiplication or division.
• Simplifying expressions involves using mathematical rules and properties.

In the given problem \(3m - 2\{m - 3[2m - (m-5)] + 4\}\), simplifying means performing operations according to established rules to condense it into \(7m + 22\). Learning to manipulate algebraic expressions enhances problem-solving skills and is foundational for higher-level math.