Problem 67 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ [-4+(-1+6)]-[7-(-6-1)] $$

Step-by-Step Solution

Answer

Question: Simplify the expression $[-4+(-1+6)]-[7-(-6-1)]$ following the order of operations. Answer: -13

Step 1: Identify the Innermost Parentheses

First, we want to locate the innermost parentheses in the expression and perform the operations inside them. Our expression is: $$ [-4+(-1+6)]-[7-(-6-1)] $$ The innermost parentheses are \((-1+6)\), and \((-6-1)\).

Step 2: Perform the Operations Inside the Innermost Parentheses

Now, we will perform the addition and subtraction inside the innermost parentheses: $$ [-1+6] = 5 \\ [-6-1] = -7 $$ So, the expression becomes: $$ [-4+(5)]-[7-(-7)] $$

Step 3: Perform the Operations Inside the Remaining Parentheses

Next, we will perform the addition and subtraction inside the remaining parentheses: $$ [-4+5] = 1 \\ [7-(-7)] = 14 $$ So, the expression becomes: $$ (1)-(14) $$

Step 4: Perform the Final Subtraction

Finally, we will perform the subtraction in the expression: $$ 1-14 = -13 $$ Hence, the simplified expression is \(-13\).

Key Concepts

Order of Operations

In mathematical calculations, especially in elementary algebra, it's crucial to follow a specific sequence when performing operations. This sequence is known as the order of operations. It makes sure that everyone solves mathematical expressions in the same way and arrives at the correct answer. The standard order of operations can be remembered with the acronym PEMDAS:

• P: Parentheses - Complete operations inside parentheses first.
• E: Exponents - Solve exponents (powers and roots, etc.).
• M/D: Multiplication and Division - Perform these operations from left to right.
• A/S: Addition and Subtraction - Again, perform from left to right.

Following this sequence ensures consistency and prevents errors. Remember, multiplication and division, along with addition and subtraction, are conducted from left to right, based on their order of appearance in the expression. Skipping this order can lead to entirely different results! Keep this sequence in mind for all your algebraic calculations.

Parentheses in Algebra

Parentheses in algebra are used to denote operations that should be completed first and provide a clear pretext that the innermost calculations should be resolved initially. They act like a highlighting tool, indicating which parts of the expression need immediate attention.

In our expression \[[-4+(-1+6)]-[7-(-6-1)]\]parentheses help to organize and prioritize calculations. We have two sets of innermost parentheses:

• (-1+6), resulting in 5.
• (-6-1), resulting in -7.

Parentheses simplify complex processes, breaking them down into manageable steps. Once calculations within them are done, they simplify the expression significantly. Without following the order suggested by parentheses, simplifying mathematical expressions can become unnecessarily complex.

Integer Operations

Integer operations involve adding, subtracting, multiplying, and dividing whole numbers. These operations play a key role in algebraic expressions like the given one. A solid grasp of integers can simplify calculations and enhance understanding of algebra.
When performing integer addition and subtraction, remember:

• Adding Negative Numbers: Similar to subtracting the absolute values. For example, adding \((-6)\) and \(-1\) results in \(-7\).
• Subtracting Negative Numbers: Equivalent to adding the absolute value. Subtracting \(-7\) in the expression \([7-(-7)]\) effectively adds \(+7\), making the operation simpler.

Operations involving integers are more straightforward than they seem. paying attention to signs and understanding basic operations can prevent mistakes and ease you through intricate problems. Ensuring each step is executed correctly leads to the correct solution.