Problem 51 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ [-17+(-4)]+[-12+15] $$

Step-by-Step Solution

Answer

The result is -18.

Step 1: Solve the first bracket

First, we solve the expression inside the first bracket: \([-17 + (-4)]\). This simplifies to \(-17 - 4\), which equals \(-21\).

Step 2: Solve the second bracket

Next, we solve the expression inside the second bracket: \([-12 + 15]\). This simplifies to \(-12 + 15\), which equals \(3\).

Step 3: Add the results from the brackets

Finally, add the results obtained from each bracket: \([-21 + 3]\). This simplifies to \(-18\).

Key Concepts

Integer Operations

Integer operations form the foundation of basic algebra. They involve performing mathematical actions with whole numbers, both positive and negative. In mathematics, integers are numbers without fraction or decimal parts. They include:

• Positive numbers: 1, 2, 3,...
• Negative numbers: -1, -2, -3,...
• Zero

It's important to handle integer operations with care as they follow specific rules based on the type of operation involved, such as addition, subtraction, multiplication, and division.
These operations become essential when dealing with equations and expressions, requiring precision and understanding of rules to simplify correctly. Remember, the sign of the numbers plays a crucial role in determining the outcome of these operations, especially in addition and subtraction.

Addition of Integers

Adding integers can seem challenging, particularly when they involve different signs. However, mastering the rules simplifies this process significantly. Let's break down some core ideas:

• When adding two positive integers, simply add their absolute values. For example, \(3 + 5 = 8\).
• When adding two negative integers, add their absolute values and keep the negative sign. For example, \((-3) + (-5) = -8\).
• When adding a positive integer and a negative integer, subtract the smaller absolute value from the larger absolute value, and take the sign of the larger absolute value. For example, \(5 + (-3) = 2\).

In our exercise, we see negative and positive numbers being added.
Follow these strategies and the rules for addition, and you'll find working with integers much easier.

Brackets in Equations

Brackets play a significant role in organizing and simplifying complex equations. They help in maintaining the structure and order of mathematical operations, often denoting which operations should be performed first. When you encounter brackets in an equation, follow these guidelines:

• Always solve the expressions inside the brackets first according to the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS/BODMAS).
• Reduce each bracketed expression to a single number before proceeding to solve any remaining parts of the equation.

In the provided exercise, two sets of brackets are used: \( [-17 + (-4)] \) and \([ -12 + 15] \).
Simplifying each bracket separately ensures that we maintain the order of operations and achieve the correct result.
Incorporating these steps into practice will enhance your ability to solve equations involving brackets confidently.