Problem 49 - Elementary and Intermediate Algebra

Question

Multiply. $$ 6(1+x) $$

Step-by-Step Solution

Answer

6 + 6x

Step 1: Apply the Distributive Property

To multiply the term outside the parentheses by each term inside the parentheses, use the distributive property. This means you will multiply 6 by 1 and 6 by x.

Step 2: Multiply 6 by 1

Calculate the product of 6 and 1. The result is 6.

Step 3: Multiply 6 by x

Calculate the product of 6 and x. The result is 6x.

Step 4: Combine the Products

Add the two products obtained from the previous steps: 6 and 6x. This gives you the final expression: 6 + 6x.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations that involves combining equal groups. You probably already know how to multiply numbers, but it's essential to understand this concept well, especially when applying it in algebra.
When you multiply two numbers, you are essentially adding one number to itself multiple times. For example, multiplying 4 by 3 means adding 4 three times (4 + 4 + 4 = 12).
In the given exercise, we multiply 6 by both terms inside the parentheses. Always remember:

• The number outside the parentheses multiplies each term inside.
• Use distributive property to handle this multiplication correctly.

Let's check the detailed method below.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. For instance, in the expression \(1 + x\), 1 is a constant, and x is a variable. Algebraic expressions are essential in algebra, as they form the building blocks for equations.
In this exercise, we work with the expression \(6(1 + x)\). Here's how to deal with such expressions:

• Constants are numbers that do not change.
• Variables are symbols used to represent unknown values, often denoted by letters like x or y.
• Operations like addition, subtraction, multiplication, and division are applied to constants and variables to form expressions.

The goal is to simplify the entire expression using basic algebra rules.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions. Terms are 'like' if they have the same variable raised to the same power. For example, \(3x\) and \(4x\) are like terms because they both contain the variable x.
Here are key points to remember when combining like terms:

• First, identify the like terms within the expression.
• Add or subtract the coefficients of these like terms.
• Keep the variable and its power the same.

In the final step of our problem, we combine the products from earlier steps. We have 6 (a constant) and \(6x\) (a term with a variable). While they can't be combined into a single term (as they are not like terms), they form the simplified expression \(6 + 6x\).