Problem 8 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ -3-2-(-5) $$

Step-by-Step Solution

Answer

The expression \(-3-2-(-5)\) simplifies to \(0\).

Step 1: Understanding the Problem

The expression given in the problem is \(-3-2-(-5)\). It contains negative numbers and a double negative operation, which can be simplified according to the rule that 'minus of negative number' becomes a positive number.

Step 2: Simplifying Double Negative

First, we simplify the double negative '-(-5)' to '+5'. So, the expression now becomes \(-3 - 2 + 5\).

Step 3: Simplifying Arithmetic Expression

Finally, we simplify the expression \(-3 - 2 + 5\) by doing the math in order. Performing the subtraction and then the addition, we get \(0\).

Key Concepts

Understanding Negative Numbers

Negative numbers are an essential part of arithmetic and represent quantities less than zero. They are the opposite of positive numbers and are usually denoted by a minus sign (-). When dealing with negative numbers in an arithmetic expression, it's crucial to recognize how they interact with other numbers. For example, subtracting a negative number is the same as adding its positive counterpart, and multiplying two negative numbers results in a positive number.

Handling negative numbers accurately is key to simplifying expressions, as they can change the result of the calculation significantly. In the given example (-3 - 2 - (-5)), it is important to interpret each negative sign correctly to understand that the last term negates the negativity of 5, effectively making it a positive number when simplifying.

Demystifying the Double Negative

A double negative occurs when two negative signs are used in succession. While in regular language usage double negatives can be confusing or considered improper, in mathematics, they serve a clear purpose. Specifically, a double negative will result in a positive. This means that in an arithmetic expression, -(-a) = a, where 'a' is any number.

In computations, recognizing and simplifying double negatives are critical steps towards finding the correct solution. The expression -3 - 2 - (-5) features a double negative with the '-(-5)'. According to the rules of arithmetic, this translates to -3 - 2 + 5. Simplifying the double negative makes it easier to proceed with and correctly solve the entire expression.

Arithmetic Expressions: Simplification Strategies

An arithmetic expression includes numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. To simplify such expressions, especially when they involve negative numbers and double negatives, applying the correct order of operations (PEMDAS/BODMAS) is crucial. However, if only addition and subtraction are present, operations are performed from left to right.

To simplify -3 - 2 + 5, as shown in the problem, you'd start by combining the first two numbers, -3 - 2, which equals -5. Then, you add 5 to this result. The collective operation simplifies to zero (0). Implementing step-by-step simplification allows for a clear and methodical route to the correct answer, making it easier to grasp and follow.