Problem 52 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ [-2+(-7)]+[-11+22] $$

Step-by-Step Solution

Answer

The sum is 2.

Step 1: Simplify the First Bracket

In the expression \[[-2 + (-7)]+[-11 + 22]\]first simplify the expression inside the first bracket. Combine \[-2\] and \[-7\] to get:\[-2 + (-7) = -2 - 7 = -9\].So, the expression becomes \[-9 + [-11 + 22]\].

Step 2: Simplify the Second Bracket

Now simplify the expression inside the second bracket. Combine \[-11\] and \[22\] to get:\[-11 + 22 = 11\].So, the expression becomes \[-9 + 11\].

Step 3: Add the Results

Finally, add the results from both brackets:\[-9 + 11\].Difference between \[11\] and \[9\] is \[2\] but since \[11\] is larger, result is positive.Thus, the result of \[-9 + 11\] is:\[2\].

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra, where complex expressions are reduced to their simplest form. Let's take a closer look at the process.
When simplifying, it is important to address one part of the expression at a time. In our original exercise, \([-2+(-7)]+[-11+22]\), we encountered two separate expressions under brackets that needed to be simplified individually. This involves combining like terms.
The first part, \([-2 + (-7)]\), was simplified by recognizing that addition of a negative number is like subtraction, so \(-2 + (-7)\) became \(-9\).
For the second part, \([-11 + 22]\), the simplification involved subtracting 11 from 22, resulting in \(11\).
By simplifying each part separately, we converted the expression to \([-9 + 11]\), making it easier to process further.

Negative Numbers

Understanding negative numbers is vital, particularly in addition and subtraction. Negative numbers are those less than zero, indicated by a negative sign (-). They behave differently from positive numbers.
When adding a negative number, it's equivalent to subtracting its positive counterpart. For example, the expression \(-2 + (-7)\) simplifies to \(-2 - 7\).
Remember that when subtracting a larger negative number, you actually increase the value. That's why \(-9 + 11\) simplifies to \(2\), because 11 is larger, making the result positive.
In performing operations with negative numbers, it's essential to keep track of the signs, as they dictate the final result's sign.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to ensure consistent results. The acronym "PEMDAS" can help you remember the order: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
In expressions like our example, \([-2+(-7)]+[-11+22]\), parentheses are handled first, simplifying what's inside each bracket before combining the results.
Each bracket operates independently, following the same rules of arithmetic clarity. This method prevents confusion and errors, ensuring that each expression is correctly simplified.
By simplifying expressions within parentheses first, and considering the entire order, we identify the final result correctly as \(2\).
Following the correct order of operations is crucial, or different and incorrect results may arise.