Problem 110 - Prealgebra and Introductory Algebra

Question

Simplify. $$-12-(-3)-(-15)$$

Step-by-Step Solution

Answer

The simplified result of the operation is 6

Step 1: Identify the numbers and their operations

The numbers to be operated on are -12, -3, and -15. The operations are subtraction for the first to second number and the second to third number.

Step 2: Remove brackets and rewrite the expression

Negative of negative becomes positive. Therefore, the expression is rewritten as: -12 + 3 + 15.

Step 3: Perform the operations

Perform the operations in sequence. Initially, -12 + 3 equals -9. Then, -9 + 15 equals 6.

Key Concepts

Simplifying Expressions

Simplifying expressions means making a mathematical expression easier to work with by removing any unnecessary components. In this exercise, the expression is \(-12 - (-3) - (-15)\). Our first task was to identify and rewrite this expression. Brackets often indicate special operations, like dealing with negative numbers.

Here's how to simplify:

• Recognize that subtracting a negative is the same as adding a positive. Therefore, \(-(-3)\) becomes \(+3\), and \(-(-15)\) becomes \(+15\).
• Rewriting the expression without these negative signs gives us: \(-12 + 3 + 15\).

This method of simplification helps in reducing errors and making calculations straightforward.

Operations with Integers

When dealing with integers, it is important to understand how different operations affect the numbers. Integers include positive and negative whole numbers, like \(-12\) and \(+15\).

Performing operations with integers follows basic rules:

• Addition: Combine values directly, and if they have different signs, subtract and keep the sign of the larger number.
• Subtraction: Consider this as adding a negative, which often requires changing signs.

In our expression, we had \(-12 + 3 + 15\). Solving involves:

• Add \(-12\) and \(+3\) to get \(-9\).
• Then, add \(-9\) and \(+15\) to get \(+6\).

Understanding these steps is key in mastering integer operations.

Negative Numbers

Negative numbers represent values less than zero. They are used to indicate opposite directions or losses. In mathematical expressions, they are handled carefully due to their effect on operations.

Key points to remember include:

• Subtracting a negative number turns into adding a positive, as demonstrated with \(-12 - (-3)\), which becomes \(-12 + 3\).
• Adding a positive number to a negative one results in finding the difference and keeping the sign of the larger number.

Working through our example, changing signs appropriately was crucial. Handling negative numbers well allows for accurate and efficient problem-solving, especially in simplifying expressions and performing operations with integers.