Solving (5+4-2) X (-2) A Step-by-Step Guide

In the realm of mathematics, we often encounter seemingly simple equations that hold a universe of underlying principles. Today, we embark on a journey to decode one such mathematical puzzle: (5+4-2) x (-2) = ? This equation, while concise, offers a gateway to understanding the fundamental order of operations and the significance of negative numbers. Let's dive into the process of solving this equation, step by meticulous step, and arrive at the correct solution.

To effectively tackle this equation, we must first grasp the crucial concept of the order of operations. This principle, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. It ensures that we arrive at a consistent and accurate answer, regardless of who is solving the equation.

In our equation, (5+4-2) x (-2) = ?, the order of operations directs us to address the expression within the parentheses first. Within the parentheses, we encounter a combination of addition and subtraction. Following the left-to-right rule for operations of equal precedence, we begin by adding 5 and 4, resulting in 9. Next, we subtract 2 from 9, which yields 7. So, the expression within the parentheses simplifies to 7.

Now, our equation transforms into 7 x (-2) = ?. We are left with a simple multiplication operation. Multiplying a positive number (7) by a negative number (-2) results in a negative product. In this case, 7 multiplied by -2 equals -14. Therefore, the solution to the equation (5+4-2) x (-2) = ? is -14.

Through this exercise, we have not only arrived at the solution but also reinforced our understanding of the order of operations and the rules governing the multiplication of signed numbers. These fundamental mathematical concepts are essential building blocks for tackling more complex equations and mathematical problems. So, the answer to our puzzle is unequivocally -14. This corresponds to option C in the provided choices. Let's delve deeper into the significance of these concepts and how they apply to various mathematical scenarios.

The order of operations, often symbolized by the acronym PEMDAS (or BODMAS in some regions), is not merely a set of arbitrary rules; it is the bedrock of mathematical consistency and accuracy. This principle ensures that when faced with an equation involving multiple operations, we follow a standardized sequence to arrive at the correct answer. Without this order, the same equation could yield different results depending on the sequence in which operations are performed, leading to mathematical chaos.

Let's dissect the components of PEMDAS to fully appreciate its significance:

• P – Parentheses (or Brackets): Operations enclosed within parentheses or brackets take precedence over all other operations. This allows us to group expressions and treat them as a single unit.
• E – Exponents: Exponents, which indicate repeated multiplication, are evaluated after parentheses. This reflects the hierarchical nature of mathematical operations, where exponentiation is a higher-order operation than multiplication or addition.
• MD – Multiplication and Division: Multiplication and division hold equal precedence and are performed from left to right. This ensures that when these operations are juxtaposed, we maintain a consistent direction of evaluation.
• AS – Addition and Subtraction: Similar to multiplication and division, addition and subtraction share equal precedence and are carried out from left to right. This left-to-right convention is crucial for maintaining accuracy when these operations are combined.

To illustrate the importance of the order of operations, consider the simple equation 2 + 3 x 4 = ?. If we were to perform the addition before the multiplication, we would get 5 x 4 = 20. However, according to PEMDAS, multiplication takes precedence, so we should first multiply 3 by 4, resulting in 12. Then, we add 2 to 12, yielding the correct answer of 14. This seemingly small difference in the order of operations leads to a significant disparity in the final result.

The order of operations is not just a mathematical technicality; it is a fundamental principle that underpins various fields, from engineering and physics to computer science and finance. In any situation where mathematical calculations are involved, adhering to the order of operations is paramount for ensuring accuracy and avoiding costly errors. By mastering this principle, we equip ourselves with a powerful tool for navigating the world of mathematics and its applications.

In the realm of mathematics, numbers are not confined to the positive spectrum. Negative numbers, those that lie to the left of zero on the number line, play a pivotal role in expanding our mathematical horizons. Understanding how negative numbers interact with positive numbers, particularly in multiplication, is crucial for solving equations and tackling real-world problems.

The rules governing the multiplication of signed numbers are elegant and consistent:

• Positive x Positive = Positive: When we multiply two positive numbers, the result is always positive. This is the most intuitive scenario, as it aligns with our everyday understanding of multiplication.
• Negative x Negative = Positive: This rule may seem counterintuitive at first, but it is a cornerstone of mathematical consistency. Multiplying two negative numbers yields a positive product. Think of it as a double negative canceling out, resulting in a positive outcome.
• Positive x Negative = Negative: When we multiply a positive number by a negative number, the result is always negative. This reflects the idea that multiplying by a negative number essentially reverses the sign of the other number.
• Negative x Positive = Negative: Similar to the previous rule, multiplying a negative number by a positive number also results in a negative product. The order of multiplication does not affect the sign of the result.

Let's illustrate these rules with some examples:

• 5 x 3 = 15 (Positive x Positive = Positive)
• (-5) x (-3) = 15 (Negative x Negative = Positive)
• 5 x (-3) = -15 (Positive x Negative = Negative)
• (-5) x 3 = -15 (Negative x Positive = Negative)

These rules are not arbitrary; they are deeply rooted in the mathematical structure of the number system. Understanding these rules allows us to confidently navigate equations involving negative numbers and arrive at accurate solutions. In our original equation, (5+4-2) x (-2) = ?, we encountered the multiplication of a positive number (7) by a negative number (-2), which resulted in a negative product (-14), as dictated by these rules.

The ability to manipulate positive and negative numbers is not just a mathematical skill; it is a valuable tool for problem-solving in various domains. From calculating financial debts and credits to understanding temperature fluctuations and electrical circuits, negative numbers provide a framework for representing quantities that exist in opposition to positive quantities. By mastering the rules of multiplication with signed integers, we unlock a deeper understanding of the mathematical world and its applications.

Now that we have delved into the underlying principles of the order of operations and the multiplication of signed numbers, let's revisit our original equation, (5+4-2) x (-2) = ?, and solidify our understanding by applying these concepts. This equation serves as a microcosm of the broader mathematical landscape, showcasing the interplay of different operations and the importance of adhering to established rules.

As we discussed earlier, the order of operations (PEMDAS) dictates that we first address the expression within the parentheses. Inside the parentheses, we have a combination of addition and subtraction: 5 + 4 - 2. Following the left-to-right rule for operations of equal precedence, we begin by adding 5 and 4, which gives us 9. Then, we subtract 2 from 9, resulting in 7. So, the expression (5+4-2) simplifies to 7.

With the parentheses resolved, our equation now reads 7 x (-2) = ?. We are left with a simple multiplication operation involving a positive number (7) and a negative number (-2). Recalling our rules for multiplying signed numbers, we know that a positive number multiplied by a negative number yields a negative product. In this case, 7 multiplied by -2 equals -14.

Therefore, the solution to the equation (5+4-2) x (-2) = ? is -14. This aligns perfectly with option C in the provided choices. By systematically applying the order of operations and the rules of signed number multiplication, we have confidently navigated this equation and arrived at the correct answer.

But the value of this exercise extends far beyond simply solving this particular equation. The principles we have employed – the order of operations and the rules for multiplying signed numbers – are universally applicable across a vast range of mathematical problems. Whether we are dealing with algebraic expressions, geometric calculations, or statistical analyses, these fundamental concepts serve as our compass, guiding us towards accurate solutions.

To further illustrate this point, let's consider a slightly more complex equation: (3 x 2 – 1) x (-4 + 2) = ?. In this equation, we encounter multiple sets of parentheses and a combination of multiplication, subtraction, and addition. However, by faithfully adhering to the order of operations, we can systematically unravel this equation.

First, we address the expressions within the parentheses. In the first set of parentheses, we have 3 x 2 – 1. Multiplication takes precedence, so we first multiply 3 by 2, resulting in 6. Then, we subtract 1 from 6, yielding 5. So, the first set of parentheses simplifies to 5.

In the second set of parentheses, we have -4 + 2. Adding a negative number is equivalent to subtraction, so this expression is the same as 2 - 4, which equals -2. Thus, the second set of parentheses simplifies to -2.

Now, our equation becomes 5 x (-2) = ?. We are left with a simple multiplication of a positive number (5) by a negative number (-2). Applying our rules for signed number multiplication, we know that the product will be negative. 5 multiplied by -2 equals -10. Therefore, the solution to the equation (3 x 2 – 1) x (-4 + 2) = ? is -10.

Through these examples, we have demonstrated the power and versatility of the order of operations and the rules for multiplying signed numbers. These concepts are not just abstract mathematical principles; they are practical tools that empower us to solve a wide array of problems. By mastering these fundamentals, we lay a strong foundation for success in mathematics and its applications.

In conclusion, the seemingly simple equation (5+4-2) x (-2) = ? has served as a gateway to explore fundamental mathematical concepts. Through this journey, we have reaffirmed the critical importance of the order of operations (PEMDAS) in ensuring accuracy and consistency in mathematical calculations. We have also delved into the intricacies of multiplying signed numbers, understanding the rules that govern the interaction between positive and negative integers.

The solution to our equation, -14, is not just a numerical answer; it is a testament to the power of mathematical precision. By adhering to established principles and following a systematic approach, we have confidently navigated the equation and arrived at the correct result. This process highlights the beauty of mathematics – a discipline built upon logic, consistency, and the pursuit of truth.

The concepts we have explored extend far beyond the confines of this specific equation. The order of operations and the rules for multiplying signed numbers are foundational elements of mathematics, applicable across various domains, from algebra and geometry to calculus and statistics. By mastering these fundamentals, we equip ourselves with the tools necessary to tackle complex problems and unlock deeper mathematical understanding.

Mathematics is not merely a collection of formulas and equations; it is a language, a way of thinking, and a framework for understanding the world around us. By embracing mathematical principles and cultivating a problem-solving mindset, we empower ourselves to make informed decisions, analyze data effectively, and appreciate the elegance of logical reasoning.

As we conclude our exploration of this mathematical puzzle, let us carry forward the lessons learned. Let us continue to embrace the challenge of mathematical exploration, seeking to deepen our understanding and appreciate the beauty of this powerful discipline. The world of mathematics is vast and intricate, but with a solid foundation and a curious spirit, we can unlock its mysteries and harness its potential.