Problem 41 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -6+[6+(-9)] $$

Step-by-Step Solution

Answer

-9

Step 1: Simplify the expression inside the brackets

First, simplify the expression inside the square brackets: \[6 + (-9)\] Since adding a negative number is the same as subtracting, this simplifies to \[6 - 9 = -3\]

Step 2: Combine the results

Now substitute \(-3\) back into the original expression: \[-6 + (-3)\] Combine the numbers to get: \[-6 - 3 = -9\]

Key Concepts

Integer Operations

Integer operations are the basic calculations we perform with whole numbers, which can be positive or negative. There are four fundamental operations: addition, subtraction, multiplication, and division.

In this exercise, we focus mainly on addition and subtraction. Let's look at the problem step-by-step:

• Addition: When adding two integers, their signs will determine the result. If both integers have the same sign, you add their absolute values, and the result retains the common sign.



• Subtraction: Subtraction can be viewed as adding the opposite of the number. For instance, subtracting a positive number is the same as adding a negative number.

Understanding these operations is crucial. It helps us tackle more complex expressions and equations without confusion.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This helps us understand and solve them more easily. In the given exercise, we simplified the expression inside the brackets first.

• Step 1: Simplify inside the brackets:

We started with \[ 6 + (-9) \]. Adding a negative number is just like subtracting its positive counterpart: \[ 6 - 9 = -3 \]. We simplify inside the brackets first because of the order of operations (PEMDAS/BODMAS).

• Step 2: Combine the results:

We then substituted \[ -3 \] back into the original problem, making it \[ -6 + (-3) \]. Combining these numbers gave us \[ -6 - 3 = -9 \].

The key to simplifying expressions is understanding how to deal with operations step by step. Use brackets to keep parts of the problem organized, making each step more manageable.

Negative Numbers

Negative numbers are just like positive numbers, but with a minus sign in front. They represent values less than zero. Handling negative numbers can be tricky, but with practice and clear rules, it gets easier.

• Addition and Subtraction: When adding two negative numbers, you add their absolute values and keep the negative sign. When subtracting a negative number, it converts to addition; for example, \[a - (-b) = a + b \].



• Practical Example: Consider the exercise we studied: \[ -6 + (-9) \]. Here, both numbers are negative. Adding their absolute values (6 and 9) gives 15, and since both were negative, the result is \[ -15 \].

Mastering negative numbers requires careful attention to signs. Always pay attention to whether you are adding or subtracting, and remember the simple rules about how negative signs interact. Regular practice helps build confidence and accuracy.