Problem 3 - College Algebra with Modeling and Visualization

Question

Apply the distributive property to \(4-(5-4 x)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-1 + 4x\).

Step 1: Simplify Inside the Parentheses

First, look at the expression inside the parentheses: \(5 - 4x\). Since there is no simplification needed, move on to the next step.

Step 2: Eliminate the Negative Sign

Rewrite the expression using the distributive property for subtraction: \(4 - (5 - 4x)\) is equivalent to \(4 - 5 + 4x\). This step involves changing the subtraction of \((5 - 4x)\) into the addition of each component reversed.

Step 3: Simplify the Expression

Combine all the like terms: \(4 - 5 + 4x\). Simplify this to get \(-1 + 4x\).

Key Concepts

Simplification

Simplification is the process of reducing an expression to its simplest form. It involves combining like terms and performing basic arithmetic operations. In the given problem, simplification is crucial for removing unnecessary clutter and making the expression more manageable.
To simplify an expression:

• First, perform any operations inside parentheses.
• Next, follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), often remembered as PEMDAS.
• Combine like terms to reduce redundancies.

In our example, we simplified the expression step by step, combining terms to reach the final form of \(-1 + 4x\).
When simplification is applied correctly, it allows us to understand and solve equations more efficiently.

Expressions

Expressions are mathematical phrases that can include numbers, variables, and operations. They represent a value but do not have an equality sign that separates them into a left and right side.
Expressions can be as simple as a single number or variable, like "4" or "x," or more complex, like \(4 - (5 - 4x)\).
In the given problem, we started with a complex expression and used the distributive property to transform it into a more straightforward one.
Working with expressions often involves:

• Identifying the components (numbers, variables, operators) within the expression.
• Transforming the expression using mathematical properties like distributive, associative, or commutative laws.
• Ensuring any simplifications or calculations adhere strictly to mathematical rules.

The objective is to manipulate expressions so that they become as simple and clear as possible. This aids in solving problems effectively.

Like Terms

Like terms are terms in an expression that have the same variable raised to the same power. They can be combined to simplify expressions, making solving equations easier.
For example, \(4x\) and \(-2x\) are like terms because they both contain the variable \(x\) raised to the first power.
On the contrary, \(4x\) and \(4x^2\) are not like terms as they have different powers of \(x\).
In our original exercise, once we applied the distributive property, we needed to identify and combine like terms:

• \(4\) and \(-5\) were the constant terms.
• \(4x\) remained as it was, as there were no other similar terms to combine it with.

By combining these like terms, we streamlined the expression to get \(-1 + 4x\).
This process of identification and combination of like terms simplifies the equation, making further calculations more straightforward.