Problem 68 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ [2-(-6+10)]-[1-(2-11)] $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$-12$$.

Step 1: Identify the innermost parentheses

First, we need to look for the innermost parentheses in the expression and simplify them. In this case, we have two innermost parentheses: $$(-6+10)$$ and $$(2-11)$$.

Step 2: Simplify the innermost parentheses

Now, we'll simplify the innermost parentheses, starting with the first one: $$-6+10 = 4$$. Then, simplify the second one: $$2-11=-9$$. Our expression should now look like this: $$[2-(4)]-[1-(-9)]$$.

Step 3: Simplify the remaining parentheses

Next, we will simplify the expression inside the remaining parentheses. For the first one, we have: $$2-4=-2$$ and for the second one, we have: $$1-(-9)=1+9=10$$. Our expression is now: $$-2-10$$.

Step 4: Perform the final operation

Finally, we have a simple subtraction left to perform: $$-2-10=-12$$. So, our simplified expression is $$-12$$.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. It is the sequence in which operations should be performed to accurately calculate an expression. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guiding principle helps you decide which part of the expression to simplify first.

• Start with any calculations inside parentheses.
• Proceed to exponents, if there are any.
• Continue with multiplication or division, from left to right.
• Finish with addition or subtraction, also from left to right.

In the example, we begin by simplifying the expressions within the parentheses before proceeding to the subtraction operations, following the PEMDAS rule.

Parentheses in Algebra

The use of parentheses in algebra serves to organize and prioritize operations within an algebraic expression. They act like mathematical 'containers' telling you to deal with the enclosed numbers or expressions first before anything outside of them. When simplifying, always start with the innermost parentheses and work your way outwards. This is demonstrated in the example where \( (-6+10) \) and \( (2-11) \) were the first expressions to be simplified. After simplifying the contents, you can remove the parentheses and continue with the rest of the problem.
Remember, the rules of signs apply when dealing with parentheses: subtracting a negative is the same as adding a positive, so \( 1-(-9) = 1+9 \).

Elementary Algebra

At its core, elementary algebra involves the manipulation of algebraic expressions and solving equations. Simplifying expressions, like the one in the example, is a fundamental skill in algebra. It involves combining like terms, using distributive properties, and understanding how to work with variables and constants.
In the given problem, algebraic simplification is done by reducing the expression into as few terms as possible, achieving the result of \( -12 \). This understanding of elementary algebra lays the foundation for more complex topics, such as solving for unknowns and working with quadratic equations.

Negative Numbers

Working with negative numbers is often a source of confusion but is essential in algebra. A negative number represents a value less than zero and appears with a minus sign (\( - \)) in front of it. When simplifying expressions involving negatives, keep in mind two basic rules:

• Subtracting a negative number is the same as addition (\( -(-x) = +x \)).
• Adding two negative numbers makes a more negative number (\( -x + (-y) = -(x+y) \)).

The given exercise had two instances of dealing with negative numbers: simplifying inside the parentheses (turning subtraction into addition) and the final operation, where two negatives led to a more negative result (\( -2 - 10 = -12 \)).