Problem 59 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$[5+(-8)]+[3+(-11)]$$

Step-by-Step Solution

Answer

The simplified expression is \(-11\).

Step 1: Simplify the First Bracket

In the expression \([5 + (-8)] + [3 + (-11)]\), start by simplifying the first bracket \(5 + (-8)\). This means subtracting 8 from 5, which equals \(-3\).

Step 2: Simplify the Second Bracket

Next, simplify the second bracket \([3 + (-11)]\), which means subtracting 11 from 3. This results in \(-8\).

Step 3: Add Results from Both Brackets

Now, add the results from both simplified expressions: \(-3 + (-8)\). This is equivalent to adding two negative numbers, resulting in \(-11\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key foundation in algebra. It involves making an expression as simple as possible, typically by performing basic arithmetic operations. The goal is to make the expression easier to handle or solve.

The expression given \( [5+(-8)]+[3+(-11)] \) requires simplification by using the rules of arithmetic operations:

• Begin by focusing on any numbers inside brackets, simplifying each set of parentheses separately.
• Perform operations such as addition or subtraction between the numbers, being aware of positive and negative signs.
• Finally, combine the results using multiplication or addition as needed to fully simplify the expression.

Understanding how to simplify expressions involves recognizing these operations within any given problem and applying them accordingly. This sets the stage for more complex mathematical operations.

Negative Numbers

Negative numbers are numbers less than zero. Understanding them is crucial when simplifying expressions, especially when they appear inside brackets or are involved in addition and subtraction. Here are some basic properties:

• A negative number is indicated by a minus sign in front of it, like \(-5\).
• Adding a negative number is the same as subtracting its absolute value (e.g., \(5 + (-8) = 5 - 8\)).
• Subtracting a negative number results in addition (e.g., \(5 - (-3) = 5 + 3\)).

In the solved problem, we saw negative numbers in both \(5 + (-8)\) and \(3 + (-11)\). The process involves replacing the operation with subtraction and helps tackle real-life scenarios, like managing debt in finances, where values are naturally negative.

Brackets in Mathematics

Brackets are used extensively in mathematics to dictate the order of operations. They clarify which operations should be performed first in an equation, especially when the expression involves multiple arithmetic operations:

• Operations inside brackets always take priority. Simplify what's inside the parentheses first.
• Use brackets to group parts of an expression that should be treated as a single entity, ensuring the right operations are performed.
• In our example, the brackets indicate two separate expressions to simplify before further calculation: \(5 + (-8)\) and \(3 + (-11)\).

To align with the rules of 'PEMDAS/BODMAS' (Parentheses/Brackets, Exponents, Multiplication and Division, Addition and Subtraction), always address the expressions within the brackets first to simplify an expression correctly. As seen, this minimizes mistakes and leads to accurate results.