Addition Property Of Inequality Step-by-Step Solution

Understanding inequalities is a fundamental concept in mathematics, with practical applications across various fields. In this article, we will dissect a step-by-step solution to an inequality problem, highlighting the properties that justify each step. By carefully examining the process, we can gain a deeper understanding of how to manipulate inequalities and arrive at the correct solution.

The Problem: A Detailed Look

Let's consider the inequality presented:

Step 1: $-6(x+3)+10<-2$ Step 2: $-6 x-18+10<-2$ Step 3: $-6 x-8<-2$ Step 4: $-6 x<6$ Step 5: $x>-1$

The core question is: What property justifies the work shown between step 3 and step 4? To answer this, we need to meticulously analyze the transition from step 3 to step 4. This involves identifying the operation performed and the mathematical principle that allows us to perform it while maintaining the integrity of the inequality.

Step-by-Step Breakdown

To properly grasp the solution, let’s break down each step:

• Step 1: $-6(x+3)+10<-2$

  This is the initial inequality we need to solve. It involves a variable x, constants, and the less-than symbol (<), indicating that we are looking for values of x that make the left side of the inequality strictly less than -2.

• Step 2: $-6 x-18+10<-2$

  In this step, the distributive property is applied. The -6 outside the parentheses is multiplied by both x and +3 inside the parentheses. This gives us $-6 * x = -6x$ and $-6 * 3 = -18$. The rest of the inequality remains the same. The distributive property is a cornerstone of algebraic manipulation, allowing us to simplify expressions by multiplying a factor across terms within parentheses. It’s crucial for expanding expressions and preparing them for further simplification. Understanding the distributive property is essential not just for solving inequalities, but for a wide range of algebraic problems, making it a fundamental concept in mathematics. It acts as a bridge, allowing us to transform complex expressions into simpler, manageable forms. The correct application of the distributive property is paramount for achieving accurate solutions in algebra. It's not just a rule to memorize, but a tool to understand and wield effectively.

• Step 3: $-6 x-8<-2$

  Here, simplification occurs by combining the constant terms. We have -18 + 10, which equals -8. The inequality is now in a more simplified form, making it easier to isolate the variable term. Combining like terms is a fundamental technique in algebra, streamlining expressions and bringing clarity to equations and inequalities. In this step, the combination of -18 and +10 to yield -8 simplifies the inequality, making the next steps clearer. This process is not merely about arithmetic; it's about organizing and simplifying mathematical expressions to reveal their underlying structure. By combining like terms, we reduce complexity and pave the way for isolating the variable, which is the ultimate goal in solving equations and inequalities. This step showcases how simplifying expressions is an integral part of mathematical problem-solving, enhancing both accuracy and efficiency.

• Step 4: $-6 x<6$

  This is the step we are focusing on. To get from Step 3 to Step 4, we added 8 to both sides of the inequality. This isolates the term with x on one side. The addition property of inequality is the key principle at play here. This property states that adding the same value to both sides of an inequality preserves the inequality. This is a critical concept, as it allows us to manipulate inequalities while maintaining their validity. The act of adding 8 to both sides is not arbitrary; it's a strategic move to isolate the variable term. The addition property of inequality is a fundamental rule that ensures the integrity of the solution process. It allows us to transform inequalities without altering their fundamental truth, making it an indispensable tool in solving a wide range of mathematical problems. The thoughtful application of this property is what allows us to isolate variables and move closer to finding a solution.

• Step 5: $x>-1$

  Finally, we divide both sides by -6. Remember, dividing (or multiplying) an inequality by a negative number requires flipping the inequality sign. Therefore, the “less than” sign becomes a “greater than” sign. The crucial step here involves dividing both sides by -6. The golden rule when dealing with inequalities is that multiplying or dividing by a negative number reverses the direction of the inequality. Thus, the '<' sign flips to a '>'. This reversal is not a mere technicality; it is a fundamental requirement to maintain the mathematical truth of the statement. The negative sign's impact on the order of numbers necessitates this flip. Without this understanding, solutions to inequalities would be incorrect. This final step highlights the subtle but crucial rules that govern inequality manipulation, reinforcing the need for meticulous attention to detail in mathematical problem-solving. The act of flipping the inequality sign is a testament to the nuances that make mathematics both precise and powerful.

The Justification: Addition Property of Inequality

The property that justifies the work between step 3 ($-6 x-8<-2$) and step 4 ($-6 x<6$) is the Addition Property of Inequality. This property states that if you add the same number to both sides of an inequality, the inequality remains true. In this case, we added 8 to both sides.

To illustrate, let's break it down:

• Starting with Step 3: $-6x - 8 < -2$
• Adding 8 to both sides: $-6x - 8 + 8 < -2 + 8$
• Simplifying: $-6x < 6$

This clearly demonstrates the application of the Addition Property of Inequality.

Why Not the Other Options?

It's important to understand why the other options might be incorrect:

• Transitive Property: The transitive property typically applies to equalities or relationships where if A = B and B = C, then A = C. This doesn’t directly apply to the operation between step 3 and step 4.

Conclusion

In summary, the Addition Property of Inequality is the principle that allows us to add 8 to both sides of the inequality in this problem, transitioning from $-6 x-8<-2$ to $-6 x<6$. Understanding these properties is crucial for accurately solving inequalities and other mathematical problems. The ability to identify and apply the correct properties is what transforms mathematical problems from daunting challenges into manageable tasks. Each property is a tool, and knowing when and how to use it is the key to success in mathematics. By meticulously breaking down the problem and understanding the principles at play, we not only arrive at the correct answer but also deepen our understanding of mathematical reasoning. The journey through solving this inequality is a testament to the power of clear steps, logical deductions, and the foundational properties that underpin mathematical truths. It’s a process that builds not just answers, but also a robust mathematical mindset.