Problem 17 - Intermediate Algebra

Question

In \(-(x-7),\) what does the negative sign in front of the parentheses represent?

Step-by-Step Solution

Answer

The negative sign negates the entire expression, reversing the signs of the terms inside.

Step 1: Understand the Expression

The given expression is \(- (x - 7)\). The negative sign in front of the parentheses suggests subtraction or negation of the entire expression inside the parentheses.

Step 2: Apply the Negative Sign

To understand the effect of the negative sign, distribute it across the terms inside the parentheses: \(-1 imes (x - 7)\).

Step 3: Distribute the Negative Sign

Distribute the \(-1\) across the terms in the parentheses: \(-1 imes x + (-1) imes (-7) = -x + 7\).

Step 4: Simplify the Expression

After distributing the negative sign, the expression simplifies to \(-x + 7\). This shows that the negative sign reverses the signs of each term inside.

Key Concepts

Negative Numbers

When dealing with algebra, negative numbers can initially seem daunting. However, they are quite straightforward once you understand their role. A negative number is simply a number that is less than zero.
Negative numbers represent situations like owing money or descending below sea level. In algebra, they have the power to reverse or negate values and expressions. When you see a negative sign in front of a variable or number, it means that you are subtracting or taking the opposite value.
For instance, in \(-(x-7)\), the negative sign outside the parentheses indicates that everything inside the parentheses should be subtracted or negated. This process involves changing the signs of all terms inside the parentheses when expanded or simplified.

Distributive Property

The distributive property is an essential tool in algebra that allows us to simplify expressions efficiently. It states that a number multiplied by a sum or difference is the same as multiplying each addend separately and then adding or subtracting the products. Mathematically, this can be written as: \(a(b + c) = ab + ac\).
In the context of \(-(x-7)\), applying the distributive property involves multiplying the \(-1\) by each term inside the parentheses.

• Firstly, multiplying \(-1\) by \(x\) gives \(-x\).
• Secondly, multiplying \(-1\) by \(-7\) results in \(+7\).

After distributing the negative sign, each term's sign within the parentheses is reversed. This property is particularly useful for removing parentheses and simplifying algebraic expressions, giving us \(-x + 7\) in this scenario.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves transforming an expression into its simplest form. The goal is to make expressions as concise and manageable as possible.
For example, the expression \(-(x-7)\) can be simplified using the knowledge of negative numbers and the distributive property.
Initially, you apply the negative sign, \(-1\), to each term within the parentheses: \(-x\) and \(+7\).

• Ensure each term in the parentheses is multiplied by \(-1\).
• Combine terms if possible (though in this exercise, there are no like terms to combine).

Simplified expressions are easier to work with in further calculations or algebraic manipulations. In this instance, the expression simplifies to \(-x + 7\), making it more straightforward to use or solve in equations.