Problem 47 - Prealgebra

Question

Simplify each expression. $$-4(x-2)$$

Step-by-Step Solution

Answer

The simplified expression is \(-4x + 8\).

Step 1: Identify the Expression to be Simplified

We need to simplify the algebraic expression \[-4(x-2)\] by distributing the multiplier \(-4\) across the terms inside the parentheses.

Step 2: Distribute the Multiplier

Use the distributive property, which states \(a(b+c) = ab + ac\). Apply this by distributing \(-4\) over \(x-2\). Calculate \(-4 \times x = -4x\) and \(-4 \times (-2) = 8\). Thus, the expression becomes:\[-4x + 8\]

Step 3: Combine Like Terms (if any)

Look for like terms to combine. In the expression \(-4x + 8\), the terms are not like terms (one is a variable term, and the other is a constant), so no further simplification is possible. The simplest form of the expression is:\[-4x + 8\]

Key Concepts

Distributive Property

When simplifying expressions, it's crucial to understand the distributive property. This property is a fundamental principle in algebra that helps us multiply a single term by each term inside a parenthesis. It's like delivering a multiplied effect throughout the group of terms. This can make expressions easier to handle, especially when simplifying or solving equations.

The distributive property follows the rule:

• For any numbers or expressions,\[a(b + c) = ab + ac\]

Let's take a closer look at how it works with our original expression, \[-4(x-2)\]:

• Distribute \(-4\) by multiplying it with each term inside the parentheses separately. \(-4 \times x = -4x\)
• Then, multiply \(-4\) by the next term, \(-2\). \(-4 \times (-2) = 8\)

Applying the distributive property in this way, we simplify the expression to \[-4x + 8\].

This property is essential for breaking down complex expressions into simpler parts.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. It is one of the basic building blocks of algebra that allows us to express mathematical ideas in a flexible way. These expressions can be as simple as \(x + 2\), or more complex like the one we simplified, \[-4(x-2)\].

Algebraic expressions can include:

• Constants: fixed numbers, like \(-2\) in our expression
• Variables: symbols that represent numbers, such as \(x\)
• Operators: such as +, −, ×, and ÷, which dictate how terms are combined

Understanding how to manipulate and simplify these expressions is vital in solving equations and understanding more advanced mathematical concepts. Every time you use the distributive property on an expression, you are engaging with these fundamental elements of algebra.

Like Terms

In algebra, like terms are terms that have the same variables raised to the same power. Identifying and combining like terms is an essential step in simplifying expressions. It helps reduce the expression to its simplest form.

Like terms share exactly the same variable part, whereas the coefficients can differ. For example, in the expression \(3x + 5x\), \(3x\) and \(5x\) are like terms.

In contrast, \(x\), \(x^2\), and \(y\) are not like terms as they involve different variables or powers. In our simplified expression, \(-4x + 8\), we don't have any similar terms because \(-4x\) includes a variable term with \(x\), while \(8\) is a constant. Thus, no further simplification is needed.

Recognizing like terms powerfully reduces complexity in algebraic work, making calculations more manageable and results clearer.