Problem 84 - Elementary Algebra

Question

Simplify. $$ -(34) $$

Step-by-Step Solution

Answer

The simplified form is 34.

Step 1: Understanding the Problem

We need to simplify the given expression \(-(-34)\). The negative sign in front indicates that we need to find the opposite of a negative number.

Step 2: Applying the Negative Sign

When we apply the negative sign to \(-34\), it becomes positive \(34\). This rule states that the negative of a negative number is a positive number.

Step 3: Final Simplification

Thus, the expression \(-(34)\) simplifies to \(34\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are commonly used in mathematics to represent values below a certain point, usually zero. However, they can also be used to indicate direction, such as in temperature below freezing. In algebra, negative numbers are represented with a minus sign (-) in front of the number.
Understanding negative numbers is crucial in algebra because they help describe many different mathematical situations. For example, if you have a debt of $34, you can represent this with \[-34\]. The negative sign indicates that the amount is not possessed but owed.
Negative numbers are essential not only in real life but also in academic subjects such as physics, economics, and engineering. When working with negative values, it’s important to remember the rules governing their operations to avoid errors.

Opposite of a Number

The concept of the opposite of a number is integral in algebraic simplification. The opposite of a number is essentially its reflection across zero on the number line. In other words, if you have a number \(-x\), its opposite would be \(x\), and vice versa.
To find the opposite of any given number, you simply change its sign:

• The opposite of a positive number is negative.
• The opposite of a negative number is positive.

For instance, in the provided exercise, you are given \(-\left(34\right)\). Applying the concept of opposites, turning the negative number to its opposite results in a positive 34. This simplification is a direct application of the rule that the negative of a negative is a positive.
The ability to determine the opposite of a number quickly is fundamental in solving algebraic equations and simplifying expressions.

Simplification Rules

Simplification in algebra involves recognizing and applying certain rules to make an expression easier to work with or solve. The goal is to break down complex expressions into simpler parts without changing the value.
Here are some key simplification rules, particularly relevant to negative numbers:

• Two negatives make a positive: When you multiply or divide two negative numbers, the result is a positive number. This is applicable when you have terms such as \(-(-x)\). By removing the double negative, the number becomes positive.
• Combine like terms: This involves adding or subtracting numbers with the same variable part, reducing the expression's complexity. However, this specific rule doesn't apply to the current exercise because there are no variable terms involved.

In our specific example from the exercise, the simplification involves applying the rule of the negative of a negative is a positive. This simplifies directly to 34, illustrating a basic yet vital simplification technique used extensively in algebra.