Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$4(x-5)$$

Step-by-Step Solution

Answer

The simplified expression is \(4x - 20\).

Step 1: Analyze the problem

Identify the type of problem and select the appropriate mathematical approach.

Step 2: Solve step by step

The simplified expression is \(4x - 20\)..

Step 3: Verify

Check the solution by substitution or alternative method.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. For example, in the expression \(4(x-5)\), "4" is a coefficient, "x" is the variable, and the operations inside and outside the parentheses describe the interactions between them.

• A coefficient is a numerical factor that multiplies a variable. In \(4(x-5)\), "4" is the coefficient of the expression.
• Variables are symbols that represent numbers whose values are not yet specified. In our example, "x" is the variable representing such a value.
• Understanding algebraic expressions allows us to model real-world situations using mathematical terms, facilitating calculations and problem-solving.

Algebraic expressions can vary in complexity. They can be as simple as a single variable or number, or more complicated like polynomials. Simplifying algebraic expressions is often necessary to make calculations easier or to find solutions to equations.

Parentheses

Parentheses are used in algebra to group parts of an expression that should be treated as a single unit. This is crucial for ensuring calculations are performed in the correct order. In \(4(x-5)\), parentheses indicate that the operation inside should be considered first.

• Parentheses can contain numbers, variables, or even other expressions. They help define priority in operations, overriding the usual order of operations (often remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Removing parentheses using the distributive property, as seen in algebraic manipulation, simplifies expressions by applying multiplication to each term within them.

For instance, when we apply the distributive property to \(4(x-5)\), we ensure each term inside the parentheses is multiplied by 4. This is a crucial step in simplifying the expression and eliminating the parentheses.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to a simpler or more manageable form. The goal is to make the expression easier to work with, often by combining like terms or removing parentheses.

• Using the distributive property, as shown in \(4(x-5)\), we distribute the multiplying factor (4) to each term inside the parentheses (\(x\) and \(-5\)), resulting in \(4x - 20\).
• Combining like terms is another method of simplifying. For example, if the expression had terms like \(3x\) and \(2x\), we would combine them to make \(5x\).
• Simplifying expressions can make solving equations or graphing functions more straightforward by reducing the complexity.

In essence, simplifying expressions helps in solving mathematical problems, making them more accessible and less prone to error during calculations.