Problem 46 - Prealgebra

Question

Simplify expression. \(4 x-4(2+x)\)

Step-by-Step Solution

Answer

The simplified expression is \(-8\).

Step 1: Distribute the Negative Sign

Start with the expression \(4x - 4(2 + x)\). Apply the distributive property to \(-4(2 + x)\) by multiplying \(-4\) with each term inside the parentheses. This results in \(4x - 8 - 4x\).

Step 2: Combine Like Terms

Observe the expression \(4x - 8 - 4x\). Combine like terms by canceling \(4x\) with \(-4x\). This simplifies the expression to \(-8\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that involves multiplying a single number by each term inside a set of parentheses. This operation helps distribute the number across all terms, simplifying complex expressions more easily. For example, in the expression \(4x - 4(2 + x)\), the number \(-4\) needs to be multiplied by every term inside the parentheses `(2 + x)`. This process is visualized in this way:

\[-4(2) + (-4)(x) = -8 - 4x\]

Distributing the number ensures each term inside the parentheses contributes to the overall expression. Using this property is crucial because it breaks down equations into manageable parts. Mastering the distributive property in prealgebra lays the groundwork for more advanced mathematics.

Combining Like Terms

After using the distributive property, the next step in simplifying expressions is to combine like terms. Like terms are terms that contain the same variables raised to the same power. Only these terms can be combined through addition or subtraction.

Consider the expression \(4x - 8 - 4x\). The terms \(4x\) and \(-4x\) are like terms because they both contain \(x\) raised to the same power (which is 1).

Here's how you combine them:

• Add the coefficients of like terms: \(4 - 4 = 0\)
• The variable part stays unchanged: \(x\)

Thus, \(4x + (-4x) = 0\), which simplifies the entire expression to just \(-8\). Combining like terms reduces the complexity of expressions and is a key skill in algebra.

Prealgebra

Prealgebra serves as a foundation for all higher-level mathematics. It introduces essential concepts like the distributive property and the process of combining like terms. Understanding these concepts helps solve more complex equations in algebra, calculus, and beyond.

In prealgebra, you learn how to manipulate expressions with confidence. Take the expression from our example, \(4x - 4(2 + x)\). By applying the distributive property and then combining like terms, you simplify it to \(-8\). This simplification process teaches critical thinking and problem-solving skills.

By grasping prealgebra concepts, students develop a strong mathematical foundation. This equips them with tools to approach equations systematically, identify simplification opportunities, and improve calculation accuracy. Remember, each mathematical concept builds on the previous, making prealgebra a critical stepping stone in your math journey.