Problem 35 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ -(4 x) $$

Step-by-Step Solution

Answer

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

Step 1: Expression

$$
\text { In Exercises 33-46, simplify the expression. }
$$
$$
-(4 x)
$$

Step 2: Apply Rules

Apply appropriate mathematical rules step by step.

Step 3: Result

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

Key Concepts

Negative Sign in Algebra

Understanding how to handle a negative sign is crucial in algebra. The negative sign, written as "-", acts as a multiplier of -1. When it comes in front of parentheses containing a mathematical term or expression, it inverses the sign of each component inside.

For instance, if you have an expression like \(-(4x)\), it means you need to multiply everything inside the parentheses by -1.

• If you see a positive term inside the parentheses, the negative sign will turn it into a negative.
• Conversely, if the term inside is negative, the negative sign in front will make it positive.

Let's see how it works with \(-(4x)\):

• The positive term "4x" is inside the parentheses.
• An invisible "-1" is sitting in front of the parentheses, waiting to transform the expression.
• Multiply \( -1 imes 4x \) to get \(-4x\).

Notice how the multiplication with \(-1\) flips the sign, providing us the solution with exact reversal of the monk sign to its inside terms.

Multiplying Expressions

When multiplying expressions, you apply the multiplication operation to all involved components. This often includes numbers, variables, or a combination of both, called algebraic terms.

In our example, multiplying \(-1\) by \(4x\):

• Multiply the constant numbers first. Here, that means multiplying \(-1\) by \(4\), resulting in \(-4\).
• Next, consider the variable part, "x". Since there are no other variables to multiply it by, this remains unchanged in the expression.

This approach helps maintain accuracy, especially when handling more complex expressions.
Multiplying expressions correctly involves:

• Understanding coefficients: the numbers attached to variables.
• The rules of multiplication, such as negative with negative yields positive.
• The importance of keeping track of the sign, using principles like negative times positive equals negative.

Applying these rules makes simplifying expressions straightforward, ensuring each part of the expression is cleanly handled without error.

Algebraic Terms

Algebraic terms form the building blocks of algebraic expressions. They can be single numbers, single variables, or combinations of numbers and variables.

• A term like \(4x\) is an algebraic term, consisting of a coefficient "4" and a variable "x".
• Each term is separated by either a plus or a minus sign in an expression.
• Algebraic terms can be explicitly expressed by combining coefficients with variables, such as \(3xy\) or \(7a^{2}b\).

These terms make it possible to perform various operations in algebra, such as addition, subtraction, or multiplication.

Knowing your algebraic terms is critical because they dictate what happens during any operation:

• When multiplying or dividing terms, treat the coefficients separately from the variables.
• Applying operations like adding or subtracting requires like terms – those having the same variable part.

Because algebra depends so heavily on manipulation of these terms, understanding them is fundamental to mastering expressions. Comprehending how they interact simplifies algebra, making it accessible to anyone who wants to learn.