Problem 55 - Algebra A Combined Function

Question

Simplify each expression. \(-2+[(8-11)-(-2-9)]\)

Step-by-Step Solution

Answer

The expression simplifies to 6.

Step 1: Parentheses First

Start by solving the expression inside the large bracket. The inner expression is \((8 - 11) - (-2 - 9)\).

Step 2: Simplify First Parentheses

Calculate \(8 - 11\). This simplifies to \(-3\).

Step 3: Simplify Nested Parentheses

Move to the nested term \(-2 - 9\). This equals \(-11\).

Step 4: Distribute the Negative Sign

Solve the expression by subtracting the nested result from the first. Change the subtraction to addition for easier calculation: \(-3 - (-11) = -3 + 11\).

Step 5: Complete the Brackets

Compute \(-3 + 11\) which equals \(8\). Now replace the whole bracketed expression with \(8\).

Step 6: Final Step

Replace the bracketed expression into the original equation: \(-2 + 8\). Simplify it to get \(6\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. They play a fundamental role in algebra, as they allow us to formulate equations and perform calculations to find unknown values.
When you simplify an algebraic expression, you reduce it to its simplest form. This involves combining like terms, removing parentheses through distribution, and reducing complex numerical expressions to obtained simplified answers. For instance, in the expression \(-2+[(8-11)-(-2-9)]\), it includes numbers, brackets, and subtraction operations.
Understanding how to manipulate and simplify these expressions helps in solving equations and is integral to mastering algebra.

Order of Operations

The Order of Operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. To remember these rules, you can use the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right).
Applying these rules ensures correct calculations. In \(-2+[(8-11)-(-2-9)]\), the first step is to simplify expressions inside the parentheses. This highlights the importance of tackling expressions like \((8-11)\) and \((-2-9)\) before moving forward.

• First, calculate expressions within brackets.
• Second, handle multiplication or division next, if any were present.
• Finally, approach addition and subtraction.

Following the order of operations is essential for achieving accuracy in mathematical problems.

Negative Numbers

Working with negative numbers involves understanding that these numbers are less than zero, represented on a number line to the left of zero. In expressions, negative numbers significantly affect the outcome.
When performing operations with negative numbers, remember:

• Adding a negative number is equivalent to subtraction, for example, \(8 + (-3) = 8 - 3\).
• Subtracting a negative number changes to addition, such as in \(-3 - (-11) = -3 + 11\).
• Multiplying or dividing two negative numbers results in a positive product or quotient, while multiplying or dividing a positive number with a negative number yields a negative result.

In the expression \(-2+[(8-11)-(-2-9)]\), recognizing these rules allows for swift and accurate simplification. Proper manipulation of negative numbers is crucial for correctness in mathematical expressions and equations.