Problem 84 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression by removing parentheses and brackets. $$5[2-(y+3)]$$

Step-by-Step Solution

Answer

The simplified expression is \(-5y - 5\)

Step 1: Remove inner brackets

The first step is to remove the inner brackets by changing the sign of the elements inside the brackets due to the negative sign before them. Hence, it will be as follows: \(5[2 - y - 3]\)

Step 2: Simplify the expression inside the brackets

Next, simplify the expression inside the brackets by performing arithmetic operation: \(5[-y - 1]\)

Step 3: Distribute the multiplication

Apply the distributive property of multiplication over subtraction, distribute 5 to -y and -1: \( -5y - 5\)

Key Concepts

Simplification

Simplification is the process of making an expression easier to work with. It involves combining like terms, reducing fractions, and performing arithmetic operations. The goal is to rewrite the expression in a simpler, more manageable form.
In algebraic expressions, simplification can also involve removing parentheses and reorganizing terms to eliminate unnecessary complexity.

• Remove parentheses: This often involves applying the distributive property and combining like terms.
• Reorganize and reduce: Look for opportunities to combine terms, such as summing constants or combining terms with the same variable.

When you look at the expression $5[2-(y+3)]$, the simplification process begins by addressing the parentheses to streamline the structure of the expression. This results in a simpler expression that can be evaluated or manipulated more easily.

Distributive Property

The distributive property is a helpful operation in algebra that lets you multiply a single term by each term inside a set of parentheses.
This technique is particularly useful when you have to remove brackets or simplify expressions.
In mathematical terms, it can be expressed as:

• A(B + C) = AB + AC
• A(B - C) = AB - AC

For example, in the problem $5[2-(y+3)]$, the distributive property is used to distribute the number 5 across the expression inside the brackets. This means multiplying 5 with each term inside the bracket, which in this case would make the expression $5(-y -1)$ turn into $-5y - 5$.
The distributive property greatly simplifies algebraic expressions by eliminating parentheses and helping combine terms.

Arithmetic Operations

Arithmetic operations are the basic operations used in mathematics: addition, subtraction, multiplication, and division.
They form the foundation on which algebra is built and are critical for simplifying expressions.

• Addition (+): Combining numbers or variables together.
• Subtraction (−): Determining the difference between numbers or terms.
• Multiplication (×): Scaling numbers or terms by another quantity.
• Division (÷): Splitting numbers or terms into equal parts.

In the example \(5(-y - 1)\), arithmetic operations are used after distributing the 5. This operation involves understanding the signs of numbers and variables. The minus sign in the original problem flips the sign of each term inside the bracket, turning it to \(5(-y - 1)\).
Then, simple multiplication finds each component of the final expression by performing \(5 \times -y = -5y\) and \(5 \times -1 = -5\). Arithmetic operations allow for simplification to occur throughout an expression, making complex calculations simpler and more straightforward.