Problem 28 - Intermediate Algebra

Question

Complete each statement so that the indicated property is illustrated. See Example 1. $-4(x-2)= ______ Distributive property and simplifying

Step-by-Step Solution

Answer

-4(x - 2) = -4x + 8

Step 1: Apply the Distributive Property

To illustrate the Distributive Property, apply the multiplication of -4 to each term inside the parentheses. This results in the expression \[-4(x - 2) = -4(x) + (-4)(-2)\].

Step 2: Simplify the Expression

Simplify each term obtained in Step 1. Multiply -4 by x to get -4x, and multiply -4 by -2 to get +8. This gives the simplified expression \[-4x + 8\].

Step 3: Combine Results

Combine the simplified terms to write the complete expression. The expression \[-4(x - 2)\] using the Distributive Property and simplification is \[-4x + 8\].

Key Concepts

Simplifying Expressions

When we simplify expressions, we tidy up algebraic statements by combining like terms and performing basic arithmetic operations to make them clearer and more straightforward.
Think of it as cleaning up a messy room; we want our expressions to look neat and simple. For the expression \[-4(x - 2)\],applying the distributive property helps us "open up" the brackets first. We distribute the factor outside the parentheses (in this case, \(-4\)) to each term inside the parentheses.
After expanding the terms, they turn into individual parts that are easier to simplify or solve. We are then able to apply arithmetic operations such as addition, subtraction, multiplication, or division.
In this step-by-step solution, after distributing the \(-4\),each term becomes cleaner and can be further simplified into a compact expression:\[-4x + 8\].Keep in mind, simplifying doesn’t change the value of the expression; it just makes it easier to work with.

Algebraic Operations

Algebraic operations are the building blocks of simplifying expressions because they involve manipulating expressions using basic arithmetic concepts. The main algebraic operations include:

• Addition
• Subtraction
• Multiplication
• Division

In our context, multiplying is the key operation since we are employing the distributive property to "spread" the multiplier across each term within the parenthesis.

For the expression\(-4(x - 2)\), the multiplication step separates each term, like so:\(-4 \cdot x\) and\(-4 \cdot (-2)\).
Once separated, these terms can either be added or subtracted if needed, but in this specific example, the multiplication finishes the task to match its given expression format. Understanding the various algebraic operations allows us to rearrange and simplify terms effectively within expressions.

Negative Multiplication

Negative multiplication adds an additional layer of complication to expressions but follows a consistent set of rules that are easy to remember.
When multiplying two numbers or terms where at least one of them is negative, pay careful attention to the sign of the result:

• If both numbers are negative, their product is positive. For example, \((-4) \cdot (-2) = 8\).
• If one of the numbers is negative, the result is negative, such as \((-4) \cdot x = -4x\).

In \(-4(x - 2)\), multiply through using this rule to ensure signs are correct for each term:\[-4 \cdot x\] gives \(-4x\),and\[-4 \cdot (-2)\]yields\(+8\).
These principles of sign change are crucial when working with expressions because any sign mistake can lead you astray in your calculations!