Problem 58 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-6-(-3)+8-11$$

Step-by-Step Solution

Answer

The simplified value of the series of additions and subtractions is -6.

Step 1: Parentheses

Start by handling the subtraction of the negative number -3. In mathematics, subtracting a negative number turns into addition, meaning -(-3) should be interpreted as +3.

Step 2: Simplify

Now, rewrite the expression by replacing '-(-3)' with '+3'. The new expression is -6+3+8-11.

Step 3: Perform the Operations

Finally, calculate the operation. Do so by following the order of operations, adding and subtracting from left to right: -6+3 equals -3. Then, -3+8 equals 5. Finally, 5-11 equals -6.

Key Concepts

Order of Operations

In algebra, the order of operations is a set of rules that governs the sequence in which calculations are performed. These rules are essential to ensure that mathematics is consistent. When faced with expressions involving multiple operations like addition, subtraction, multiplication, and division, following the order of operations ensures that we obtain the correct result.

A common mnemonic for remembering the order is "PEMDAS," which stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

In our exercise, we're focused on addition and subtraction. The rule is to perform these operations from left to right.
Ordering the operations correctly is crucial, as doing them out of order can lead to incorrect answers. By following these rules, you can simplify complex operations accurately and efficiently.

Integer Arithmetic

Integer arithmetic is the branch of arithmetic dealing with whole numbers, both positive and negative. In our exercise, we are dealing with integer operations involving addition and subtraction. Often, confusion arises with operations like subtracting a negative number, but with some understanding, it becomes straightforward.

Here are some key points to remember:

• Negative and positive values are treated differently, and their operations need careful attention.
• Subtracting a negative integer is the same as adding its positive counterpart. For instance, \(-(-3)\) is equivalent to \(+3\).
• Add or subtract in the order as they appear in the expression when no parentheses guide otherwise.

Through practice, working with integers becomes easy, allowing you to handle more complex expressions with confidence.

Simplification Steps

Simplification in algebra involves reducing an expression to its simplest form while retaining its equivalence. The goal is to make calculations easier and reduce the chances for mistakes. In our example, we carry out simplification by performing the operations in a step-by-step manner, adhering to the rules.

When simplifying the expression \(-6 - (-3) + 8 - 11\):

• First, recognize that \(-(-3)\) becomes \(+3\), by the rule that subtracting a negative turns into addition.
• Next, rewrite the expression as \(-6 + 3 + 8 - 11\).
• Then, compute sequentially left to right: \(-6 + 3 = -3\), \(-3 + 8 = 5\), and finally \(5 - 11 = -6\).

Each of these simplification steps is necessary to reach the final and correct answer. By consistently applying these steps, students can solve similar algebraic problems successfully.