Simplifying The Expression (3-8) + (-6) A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex or seemingly convoluted mathematical statements and reduce them to their most basic and understandable forms. This process not only makes the expressions easier to work with but also unveils their underlying meaning and relationships. In this article, we will delve into the step-by-step simplification of the expression (3-8) + (-6). This example serves as a practical illustration of how to apply the order of operations and the rules of signed numbers to arrive at a concise and accurate solution. Whether you are a student grappling with basic arithmetic or a seasoned mathematician seeking a refresher, this comprehensive guide will provide a clear and insightful understanding of the simplification process.

Understanding the Order of Operations

When simplifying mathematical expressions, adhering to the order of operations is paramount. This established convention ensures that everyone arrives at the same correct answer. The order of operations is often remembered by the acronym PEMDAS, which stands for:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

The order of operations dictates the sequence in which we perform mathematical operations. Parentheses take precedence, followed by exponents, then multiplication and division (performed from left to right), and finally, addition and subtraction (also performed from left to right). By adhering to this order, we eliminate ambiguity and ensure consistent results. For instance, in an expression like 2 + 3 * 4, we would perform the multiplication (3 * 4) before the addition, yielding 2 + 12 = 14, rather than incorrectly adding 2 and 3 first. Grasping and applying the order of operations is not just a procedural necessity; it is the bedrock of accurate mathematical manipulation. It is the guiding principle that allows us to dissect complex expressions and reconstruct them in a simplified, understandable form.

Step-by-Step Simplification of (3-8) + (-6)

Let's embark on the simplification journey of the expression (3-8) + (-6), meticulously dissecting each step to ensure clarity and understanding.

Step 1: Simplify within the Parentheses

Our initial focus is on the expression enclosed within the parentheses: (3-8). This is the first priority according to the order of operations. Subtracting 8 from 3 results in a negative number. To visualize this, consider a number line. Starting at 3 and moving 8 units to the left, we land at -5. Therefore, (3-8) simplifies to -5.

Step 2: Rewrite the Expression

Having simplified the parentheses, we now rewrite the original expression, replacing (3-8) with its simplified form, -5. This gives us a new, intermediate expression: -5 + (-6).

Step 3: Adding Signed Numbers

In this step, we encounter the addition of two negative numbers: -5 and -6. When adding numbers with the same sign (in this case, both are negative), we add their absolute values and retain the sign. The absolute value of a number is its distance from zero, regardless of direction. Thus, the absolute value of -5 is 5, and the absolute value of -6 is 6. Adding these absolute values gives us 5 + 6 = 11. Since both original numbers were negative, the result is also negative. Therefore, -5 + (-6) equals -11.

Step 4: Final Simplified Expression

After performing the addition, we arrive at the final simplified form of the expression. The expression -5 + (-6) simplifies to -11. This is the most concise and understandable representation of the original expression (3-8) + (-6).

In summary, the simplification process involved first addressing the parentheses, then rewriting the expression, and finally, adding the signed numbers. By meticulously following the order of operations and the rules of signed numbers, we successfully transformed the initial expression into its simplified form, -11. This step-by-step approach underscores the importance of methodical calculation in mathematics, ensuring accuracy and clarity in our results.

The Importance of Signed Numbers

Understanding signed numbers, also known as integers, is crucial for simplifying expressions effectively. Signed numbers encompass both positive and negative numbers, including zero. The sign of a number indicates its direction relative to zero on a number line. Positive numbers are greater than zero and extend to the right, while negative numbers are less than zero and extend to the left. Mastering the rules for operating with signed numbers is essential for accurate mathematical calculations.

Adding Signed Numbers

When adding signed numbers, the rules depend on whether the numbers have the same sign or different signs.

  • Same Signs: If the numbers have the same sign (both positive or both negative), we add their absolute values and retain the sign. For example, 3 + 5 = 8 (both positive, result is positive) and -2 + (-4) = -6 (both negative, result is negative).
  • Different Signs: If the numbers have different signs (one positive and one negative), we subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. For example, -7 + 4 = -3 (absolute values are 7 and 4, 7 is larger, so the result is negative) and 9 + (-2) = 7 (absolute values are 9 and 2, 9 is larger, so the result is positive).

Subtracting Signed Numbers

Subtraction of signed numbers can be thought of as adding the opposite. To subtract b from a, we add the additive inverse of b (which is -b) to a. In other words, a - b is equivalent to a + (-b). This transformation simplifies subtraction into an addition problem. For instance, 5 - 8 is the same as 5 + (-8), which equals -3. Similarly, -3 - (-2) is equivalent to -3 + 2, resulting in -1. Understanding this principle allows us to apply the rules of addition to subtraction problems, streamlining calculations and reducing errors.

Multiplying and Dividing Signed Numbers

The rules for multiplying and dividing signed numbers are straightforward and consistent.

  • Same Signs: When multiplying or dividing two numbers with the same sign (both positive or both negative), the result is positive. For example, 4 * 3 = 12 and (-2) * (-5) = 10. Similarly, 10 / 2 = 5 and (-8) / (-4) = 2.
  • Different Signs: When multiplying or dividing two numbers with different signs (one positive and one negative), the result is negative. For example, 6 * (-2) = -12 and (-9) * 3 = -27. Likewise, (-15) / 3 = -5 and 20 / (-4) = -5.

These rules for signed numbers are fundamental in algebra and beyond. Mastery of these rules ensures accuracy in simplifying expressions, solving equations, and tackling more advanced mathematical concepts. The ability to confidently navigate the world of positive and negative numbers is a cornerstone of mathematical proficiency.

Common Mistakes to Avoid

When simplifying expressions, it's easy to make errors if we're not careful. To ensure accuracy, let's examine some common pitfalls and how to avoid them.

Neglecting the Order of Operations

One of the most frequent errors is disregarding the order of operations. As we discussed earlier, PEMDAS dictates the sequence in which operations must be performed. Failing to adhere to this order can lead to incorrect results. For instance, consider the expression 8 + 2 * 3. If we mistakenly add 8 and 2 first, we get 10 * 3 = 30, which is wrong. The correct approach is to multiply 2 and 3 first, resulting in 8 + 6 = 14. Always remember PEMDAS to prevent this common error. To reinforce this concept, practice with various expressions, consciously applying the order of operations until it becomes second nature.

Sign Errors

Sign errors are another common source of mistakes, particularly when dealing with negative numbers. These errors often occur during addition, subtraction, multiplication, or division. For example, incorrectly subtracting a negative number can flip the sign, leading to a wrong answer. Remember that subtracting a negative is the same as adding a positive. A helpful strategy is to rewrite subtraction problems as addition problems by changing the sign of the number being subtracted. Careful attention to signs is crucial, and double-checking each step can significantly reduce these errors. Additionally, using a number line can provide a visual aid to confirm the correctness of operations involving signed numbers.

Incorrect Distribution

Distribution is a powerful tool for simplifying expressions, but it must be applied correctly. The distributive property states that a(b + c) = ab + ac. An error occurs when we fail to distribute across all terms within the parentheses. For example, in the expression 2(x + 3), we must multiply both x and 3 by 2. The correct result is 2x + 6, not 2x + 3. To avoid this, systematically distribute the term outside the parentheses to each term inside. It's often helpful to draw arrows to visually connect the term being distributed to each term inside the parentheses, ensuring that no term is missed.

Combining Unlike Terms

In algebraic expressions, we can only combine like terms – terms that have the same variable raised to the same power. A common mistake is to combine unlike terms, such as 2x and 3x². These terms cannot be combined because they have different powers of x. To avoid this error, carefully identify like terms before attempting to combine them. Underlining or highlighting like terms can be a useful strategy. Remember, combining unlike terms violates fundamental algebraic principles and leads to incorrect simplifications. A solid understanding of what constitutes a “like term” is essential for algebraic accuracy.

By being aware of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy in simplifying expressions. Math is like a sport, more practice you do better you get. Regular practice, combined with careful attention to detail, is the key to mathematical proficiency.

Conclusion

Simplifying mathematical expressions is an essential skill that underpins success in algebra and beyond. In this article, we meticulously simplified the expression (3-8) + (-6), illustrating the importance of the order of operations and the rules for manipulating signed numbers. We emphasized the step-by-step approach, highlighting the initial simplification within parentheses, rewriting the expression, and finally, performing the addition of signed numbers to arrive at the concise solution of -11.

We also delved into the significance of understanding signed numbers, exploring the rules for addition, subtraction, multiplication, and division. Mastering these rules is critical for accurate calculations and confident mathematical manipulation. Furthermore, we addressed common mistakes to avoid, such as neglecting the order of operations, making sign errors, incorrect distribution, and combining unlike terms. By being mindful of these pitfalls and employing strategies to prevent them, we can enhance our accuracy and problem-solving abilities.

The journey of simplifying expressions is not just about finding the right answer; it is about developing a systematic and logical approach to problem-solving. Each step in the simplification process builds upon the previous one, fostering a deeper understanding of mathematical principles. This understanding empowers us to tackle more complex problems with confidence and precision.

In conclusion, whether you're a student embarking on your mathematical journey or a seasoned practitioner seeking a refresher, the principles and techniques outlined in this article serve as a valuable resource. Simplify expressions with care, adhere to the order of operations, master the rules of signed numbers, and avoid common mistakes. With practice and persistence, you'll develop the skills necessary to excel in mathematics and beyond. Remember, mathematics is not just about numbers and equations; it's about developing critical thinking and problem-solving skills that are applicable in all aspects of life.