Problem 65 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ (-6+2)-(5-11) $$

Step-by-Step Solution

Answer

Question: Simplify the expression $(-6+2)-(5-11)$. Answer: 2

Step 1: Identify the operations inside the parentheses

First, we need to identify the operations inside the parentheses and perform those calculations. $$ (-6+2) \quad \text{and} \quad (5-11) $$

Step 2: Perform operations inside the parentheses

Next, calculate the sum and difference inside the parentheses: $$ (-6+2) = -4 \quad \text{and} \quad (5-11) = -6 $$ So now the expression becomes: $$ -4-(-6) $$

Step 3: Simplify the expression

To simplify the expression, we should remember that subtracting a negative number is equivalent to adding the positive counterpart of that number: $$ -4-(-6) = -4+6 $$

Step 4: Perform the remaining operation

Finally, perform the remaining addition operation: $$ -4+6 = 2 $$ So the simplified expression is 2.

Key Concepts

Expressions

Algebraic expressions involve numbers, variables, and operations like addition, subtraction, multiplication, and division. They are mathematical phrases representing a quantity. By manipulating expressions, we solve problems related to various quantitative scenarios.
When dealing with expressions such as oence((a + b) - (c - d)),
the structure can often be layered with parentheses and different operations. Understanding and correctly parsing these expressions is crucial for carrying out the operations in the right order.

• Order of Operations: When evaluating expressions, always follow the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
• Term Identification: Look for groups separated by addition or subtraction; these are the terms of your expression. For instance, in oence((3x + 4) - (2 - x)), there are two main terms:
  (3x + 4) and oence((2 - x)).

Understanding expressions helps lay the groundwork for simplifying and solving algebraic equations.

Parentheses

Parentheses are special symbols used in mathematics to group parts of an expression. They tell you what operations are to be performed first. In our exercise, the parentheses put focus on computing the values inside before handling any other operations.
Ignoring parentheses can result in different values. For instance, in oence((-6+2)-(5-11)),
the expressions within the parentheses must be resolved first:

• Nested Computations: If there are parentheses within parentheses, compute from the innermost to outermost.
• Functions of Parentheses: Besides denoting order, parentheses can also indicate multiplication, such as oence(2(x+3)), where you multiply the term 2 by the entire contents within the parentheses.

Always respect the parentheses in expressions to ensure accurate calculations.

Simplifying Expressions

Simplifying expressions turns complex or lengthy expressions into simpler ones, while retaining their original value. It involves reducing expressions down to their most basic form.
In our problem, oence(-4 - (-6)),
simplification is performed by changing a double negative to a positive, resulting in oence(-4 + 6).
Yet, there are additional techniques involved in simplifying expressions:

• Combining Like Terms: Terms that have identical variable parts such as oence(3x and 5x) can be combined.
• Using Arithmetic Tools: Leverage properties like distributive, associative, and commutative to arrange or manipulate terms.
• Eliminating Double Negatives: Remember that the series of operations "negative of a negative" results in a positive, simplifying the computation.

Mastering the skill of simplifying helps to better understand and tackle more complex algebraic challenges.