Problem 34 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -(-1) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression -(-1) is 1.

Step 1: Interpret the problem

The expression -(-1) can be thought of as the opposite of -1. Another way to think about this is subtracting a negative number, which is the same as adding its positive counterpart.

Step 2: Change the signs

Since subtracting a negative is the same as adding a positive, we can rewrite the expression as follows: $$ -(-1) = +(+1) $$

Step 3: Simplify the expression

Now, the expression is simplified to: $$ +1 $$ So the simplified form of the given expression is 1.

Key Concepts

Integer Operations

Integer operations involve basic mathematical functions that we perform with integers, such as addition, subtraction, multiplication, and division. Integers include all whole numbers, both positive and negative, including zero.
When we perform operations using integers, keeping the signs in mind is crucial. For instance, when we add two positive numbers, we get a positive result. Similarly, adding two negative numbers results in a more negative value.

• Addition and Subtraction: When adding or subtracting, the sign determines the direction on the number line. Subtracting is equivalent to adding the opposite.
• Multiplication and Division: Multiplying or dividing same-sign numbers yields a positive result, while opposite signs yield a negative result.

Understanding integer operations well is key, not only for basic arithmetic but also for algebra operations as we progress to more complex mathematical expressions.

Negative Numbers

Negative numbers are numbers that are less than zero, represented with a minus sign (-) in front. They are crucial in mathematics for representing values below a defined point, like temperature or depth below sea level.
Working with negative numbers can initially be confusing. But there are rules to make it simple:

• A negative times a positive is negative.
• A negative times a negative is positive.
• Subtracting a negative number is the same as adding its positive counterpart.

In algebra, it’s common to see the operation of subtracting a negative. For example, \(-(-1)\) means taking the opposite of \(-1\), which results in positive 1. This concept of sign change is pivotal for simplifying expressions and solving equations.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This usually means performing operations that make the expression easier to work with or understand.
When simplifying, follow these basic strategies:

• Combine like terms: Terms that have the same variables can be combined.
• Apply arithmetic rules: Use basic operations correctly to simplify.
• Change double negatives: Double negatives in an expression (like \(-(-x)\)) are equivalent to a positive.

The initial expression \(-(-1)\) simplifies to 1 by removing the double negative. Simplifying not only makes calculations easier but also helps in understanding the underlying values or quantities expressed in the algebraic expression.