Problem 94 - Elementary Algebra

Question

Simplify. \(-(-1) 5\)

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Understand the Negative Sign

The given expression is \(-(-1)5\). First, recognize that it is equivalent to multiplying \(-1\) by \(5\) and then applying the negative sign to the \(-1\). It is key to accurately interpret the negative sign at the beginning of the expression.

Step 2: Apply the Negative Sign

Apply the negative sign in front of the parentheses \(-(-1)\). Converting \(-(-1)\) results in \(1\), since negating a negative number results in a positive number. Therefore, \(-(-1)\) simplifies to \(1\).

Step 3: Multiply

Now that the expression inside the parentheses is simplified to \(1\), the expression becomes simple multiplication: \(1 imes 5\). Multiply \(1\) by \(5\) to get \(5\).

Key Concepts

Understanding Negative Numbers

Negative numbers can seem tricky at first, but they follow simple rules. These numbers are less than zero and are often shown with a "-" sign in front, indicating their value on the opposite side of the number line compared to positive numbers. When working with negative numbers, it's important to remember:

• A negative times a negative always results in a positive. For example, \[ (-1) \times (-1) = 1 \] This is because negating a negative value means you move in the opposite direction on the number line, effectively making it positive.
• A negative times a positive results in a negative. \[ (-1) \times 5 = -5 \]

Understanding how to handle negative signs will help you simplify expressions like \(-(-1) 5\), where double negatives simplify to a positive number.

The Basics of Multiplication with Signs

Multiplication is a fundamental operation in mathematics, and when combined with signs, it can change the result drastically. Here's what you need to know:

• Signs in Multiplication: Always pay attention to the signs in front of the numbers you are multiplying. The sign rules are straightforward.
  • Multiplying two numbers with the same sign (both positive or both negative) gives a positive result. \( -(-1) \times 5 = 1 \times 5 \)
  • Multiplying numbers with different signs (one positive, one negative) yields a negative result.
• Zero in Multiplication: Remember, no matter what sign the other numbers have, multiplying by zero always gives zero.

In this specific example, after changing \(-(-1)\) to \(1\), we simply multiply \(1\) by \(5\), resulting in \(5\). This emphasizes the importance of evaluating the signs before carrying out the multiplication.

Working with Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations representing mathematical relationships. Simplifying these expressions often involves simplifying negative signs and ensuring correct operations are used. When tackling expressions:

• Break Down the Expression: Look at each part, just like we did with \(-(-1)5\). Start with handling negative signs and parentheses.
• Follow the Order of Operations: Remember PEMDAS/BODMAS rules (Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)). In our example, we first convert \(-(-1)\) before multiplying by \(5\).
• Rearrange if Necessary: Sometimes changing the order of multiplication or grouping different parts can simplify solving.

By carefully considering each part of the expression and applying the basic rules, algebraic expressions like this one become less daunting. Always take it one step at a time to ensure accuracy.