Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$2(4 x-5)$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(2(4x - 5)\) without parentheses is \(8x - 10\).

Step 1: Identify the numbers

First, identify the numbers in the given algebraic expression, \(2(4x-5)\), which are 2, 4, and -5. Also, recognize that 'x' is a variable.

Step 2: Apply the distributive property

Apply the distributive property by multiplying each term inside the parentheses by the number outside the parentheses. This means 2 is multiplied by 4x, and 2 is also multiplied by -5.

Step 3: Simplify the products

Multiply 2 and 4x to get 8x, and multiply 2 and -5 to get -10. Place the results next to each other separated by a minus sign (since there's a minus sign between 4x and 5 in the original expression). This results in the simplified algebraic expression: \(8x -10\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operators such as addition, subtraction, multiplication, or division. For example, the expression \(2(4x-5)\) is made up of several components.

• Numbers: These are constant values such as 2, 4, and -5 in our example.
• Variables: These are symbols, often represented by letters, such as the 'x' in 4x, that can take on different numerical values.

Understanding algebraic expressions is a crucial first step in solving problems because they represent real-world situations in a mathematical form. Manipulating these expressions—like using the distributive property to rewrite \(2(4x-5)\) as \(8x - 10\)—allows us to simplify and solve complex problems.

Variables in Algebra

Variables are foundational in algebra, represented by symbols (usually letters) that stand for unknown or changeable values. In our example, the variable is 'x'.

Here’s why variables are important:

• Flexibility: Variables allow us to construct expressions or equations that illustrate general truths or relationships, useful for many different sets of numbers.
• Problem Solving: By manipulating variables, we can find out which values make an equation true.

In the expression \(2(4x-5)\), 'x' can be any number. Algebra essentially depends on the ability to interpret and solve these expressions or equations for such variables. When you perform operations like applying the distributive property, you are breaking down complex expressions to simpler ones, often making it easier to find the value of a variable.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form without changing their value. This often entails combining like terms and using properties like the distributive property.

In the given exercise, \(2(4x-5)\), simplifying involves applying the distributive property:

• First, distribute the 2 across the terms inside the parentheses: multiply 2 with 4x and then 2 with -5.
• Perform the multiplication \(2 \times 4x = 8x\) and \(2 \times -5 = -10\).
• Combine these products to rewrite the expression as \(8x - 10\).

This simplification process makes it easier to understand and communicate the value of an expression. It helps in further problem-solving, such as in equations, where identifying solutions becomes more straightforward. The process of simplification ensures that expressions are as direct and clear as possible, avoiding any unnecessary complexity.