Problem 12 - Fundamentals of Mathematics

Question

Find the opposite of each number. \(-[-(-7)]\)

Step-by-Step Solution

Answer

The opposite is 7.

Step 1: Simplify Inner Parentheses

First, we simplify the expression inside the innermost parentheses: \(-(-7)\). The negative of \(-7\) is a positive 7. Therefore, \(-(-7) = 7\).

Step 2: Simplify Next Parentheses

Next, take the simplified result from Step 1 and simplify the next operation: \(-[7]\). The negative of 7 is \(-7\). Thus, \(-[7] = -7\).

Step 3: Final Simplification

Finally, take the result from Step 2 and simplify the expression: \(-[-7]\). The negative of \(-7\) is a positive 7, thus \(-[-7] = 7\).

Key Concepts

Integer Operations

Integer operations are basic mathematical processes involving whole numbers. These operations include addition, subtraction, multiplication, and division. When working with integers, it's essential to understand how they interact, especially when they involve negative numbers. For instance, subtracting a negative number is equivalent to adding a positive one. Similarly, multiplying or dividing two negative numbers yields a positive result.

Knowing the signs and the effects of operations is crucial for solving expressions and equations accurately. You deal with numbers that aren't whole or decimals, so comprehending these rules aids in problem-solving and mathematical fluency.

• Adding two positive integers gives a positive result.
• Adding two negative integers results in a negative answer.
• Adding a positive and a negative integer depends on their magnitudes.
• Multiplying/dividing same-signed integers results in a positive number.
• Multiplying/dividing different-signed integers results in a negative number.

Parentheses Simplification

Parentheses simplification is a fundamental concept used to determine the order in which operations are performed in algebra. Parentheses dictate which calculations you should complete first. The innermost parentheses should be simplified before addressing the outer ones.

This process helps avoid mistakes in calculation, especially in complex expressions. The basic sequence is:

• Solve what is inside the innermost parentheses first.
• Once solved, move outward to the next set of parentheses.
• Continue this process until the expression is fully simplified.

This order of operations ensures clarity and accuracy in solving mathematical expressions. For example, in the expression \(-[-(-7)]\), the innermost operation yields 7 because the negative of -7 is 7, then proceed to solve \(-[7]\), and finally, \(-[-7]\), resulting in positive 7.

Negative Sign Rules

Negative sign rules are essential when working with integers, especially within operations involving multiple negative numbers. Understanding these rules helps simplify expressions and solve problems more accurately.

When you see a negative sign outside of parentheses, it affects all the terms within the parentheses. Here’s how:

• The negative of a negative is a positive (-(-7) = 7).
• The negative of a positive is a negative ((-7) = -7).

If you multiply or divide two integers, the negative sign rules dictate the sign of the outcome. These rules are straightforward but can dramatically alter the result if misunderstood.

By carefully applying these rules to expressions with several negative signs, like in \(-[-(-7)]\), you can confidently determine the overall result without error. This knowledge helps demystify complex algebraic expressions and is vital for ensuring correctness in mathematics.