Problem 366 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 18-4(x+2) $$

Step-by-Step Solution

Answer

10 - 4x.

Step 1 - Identify the Distributive Property

The Distributive Property states that for any numbers a, b, and c: a(b + c) = ab + ac In our case, we need to apply this property to the term -4(x + 2)

Step 2 - Distribute the -4

Multiply -4 by each term inside the parentheses: -4(x + 2) = -4x - 8

Step 3 - Combine like terms

Substitute the distributed terms back into the original expression and simplify: 18 - 4(x + 2) = 18 - 4x - 8 Combine the constant terms (18 and -8): 18 - 8 = 10 So the simplified expression is: 10 - 4x

Key Concepts

Algebraic Simplification

In algebra, simplification is the process of rewriting an expression in a simpler form. This doesn't change the value, but makes it easier to work with. For example, in our given expression: 18 - 4(x + 2), we simplify it to make calculations and interpretations easier. Simplification often involves different algebraic rules and properties, such as the Distributive Property.
Simplification can be summarized in a few steps:

• Remove parentheses using the distributive property or other algebraic rules.
• Combine like terms to reduce the expression to its simplest form.
• Reorganize and rewrite the expression as needed.

Each of these steps helps make the expression easier to handle, especially when moving on to solve equations or perform further operations.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power.
For example, in the expression 18 - 4x - 8, the constants (18 and -8) are like terms because they are both just numbers without variables. To simplify, you combine these terms:

• First, identify the like terms: 18 and -8.
• Next, perform the operation: 18 - 8 = 10.

After combining like terms, our expression becomes 10 - 4x. This makes the expression shorter and easier to work with.
It’s important to remember that terms with different variables or different powers are not like terms and cannot be combined in this way. For instance, 4x and 4x^2 are not like terms.

Negative Multiplication

Multiplying negative numbers can sometimes be confusing, but there are a few rules to simplify the process. When you multiply a negative number by another expression, each part of that expression gets multiplied by the negative number.
In our example: -4(x + 2), we distribute -4 to both x and 2:

• -4 * x = -4x.
• -4 * 2 = -8.

This gives us -4x - 8. Remember, the signs are crucial to get correct. A negative times a positive is always negative, and a negative times a negative is positive (though we didn’t use this in our example).
Understanding how to handle negative multipliers helps in various aspects of algebra, ensuring accuracy while simplifying complex expressions.