Problem 32 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -(-2) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression -(-2) is 2.

Step 1: Rewrite the expression as a product

First, we will rewrite the expression as a product since the expression represents the product of two negative numbers. The expression becomes: $$ (-1)(-2) $$

Step 2: Apply the rule for multiplying two negative numbers

Now, we will apply the rule that states that multiplying two negative numbers results in a positive number. Therefore, we have: $$ (-1)(-2) = 2 $$

Step 3: Simplified expression

With the calculations completed, we now have the simplified form of the expression: $$ -(-2) = 2 $$

Key Concepts

Negative Numbers

Negative numbers are essential in mathematics and appear in various contexts such as temperature below zero, debts, or elevations below sea level. These numbers are represented with a minus sign (-) and signify a value less than zero. Understanding negative numbers includes knowing how they behave in arithmetic operations.

When you add a negative number, you essentially move to the left on the number line. For example, adding \(-3\) to \(5\) gives \(2\) because you shift three units down from five.

• Moving left for subtraction
• Moving right for addition

Handling negatives is crucial, especially when involved in subtraction or multiplication, as they often transform operations in non-intuitive ways.

Multiplication Rules

Multiplication involving negative numbers progresses beyond simple memorization. We use specific rules to determine the sign of the product.

• Positive times positive equals positive.
• Negative times negative equals positive.
• Negative times positive equals negative.
• Positive times negative equals negative.

The critical takeaway is that multiplying two numbers with the same sign, whether both positive or negative, results in a positive product. If the signs differ, you end up with a negative product.

Let's consider the product \((-1)(-2)\). Since both numbers are negative, the result is positive, hence \((-1)(-2) = 2\). This demonstrates how the rule applies to scenarios involving two negatives effectively becoming a positive.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves rewriting expressions in a more concise form. It includes reducing complexity by handling operations such as addition, subtraction, multiplication, division, and factoring out terms where applicable.

To simplify an expression like \(-(-2)\), we think of the negative sign outside the parentheses as \((-1)\) multiplied by \(-2\). This transformation lets us apply the multiplication rules directly.

• Rewrite as a product: \((-1)(-2)\).
• Multiply by the rule of negatives: \(2\).

The expression simplifies to just \(2\), revealing that two negatives "cancel out" to yield a positive. Simplification helps make complex expressions more manageable and is a cornerstone of solving equations efficiently.