Problem 69 - Prealgebra and Introductory Algebra

Question

Simplify. $$-(46)$$

Step-by-Step Solution

Answer

-46

Step 1: Understanding Negation

The negative sign in front of the parentheses means that we are going to take the opposite or negate the value within the parentheses. In simple term, changing the sign. If the number is positive, it'll become negative and if it's negative, it'll become positive. The number to be negated here is 46, which is positive.

Step 2: Apply Negation

Applying the negation to 46, the positive number becomes negative. So, we get \(-46\).

Key Concepts

Negative Numbers

When we talk about negative numbers, we're referring to numbers that are less than zero. They have a minus sign (-) in front of them. For example, -1, -5, and -46 are negative numbers. Negative numbers are interesting because they represent values less than nothing, which can be a bit tricky to understand at first.

In real life, negative numbers show up in many situations, like in temperatures below zero degrees or when you owe someone money. When you see a negative sign, think of it as going backwards on a number line. If you start at zero and move left, the numbers you pass are negative numbers.

• Example: Starting at zero, moving left by 46 units places you at -46.
• Every negative number can be understood as the *absence* or *reduction* from zero.

Opposites in Mathematics

In math, finding opposites can help us simplify and solve problems. The opposite of a given number is simply that number with the opposite sign. This means it switches from positive to negative or vice versa.

Consider the number 46. Its opposite is -46. Conversely, the opposite of -46 is 46. We often use negation to find these opposites, which helps in operations such as subtraction and when balancing equations.

Opposites are important in mathematics because they help us understand concepts like inverse operations.

• Example: The opposite (negation) helps balance equations like 46 + (-46) = 0.
• Understanding opposites is vital in solving equations where terms need cancellation.

Simplification

Simplification in mathematics involves making an expression easier to work with or understand. It can involve reducing an equation, finding the simplest form of a number, or negating a term to make calculations more straightforward.

In the exercise provided, we see simplification through negation. Simplifying \( -(46) \) means applying the negative sign, which transforms the positive number 46 to its opposite, -46. This kind of simplification helps clarify expressions involving different mathematical operations.

Simplification is about streamlining, ensuring that mathematical expressions are as clear and manageable as possible.

Simplification aims to make numbers or equations easier to understand.

Negative signs aid in simplifying by showing the opposite value.