Property Justifying Inequality Transformation

Jul 10, 2025 by ADMIN 46 views

Solving Inequalities Step-by-Step: Identifying the Property Justifying the Work

In this article, we will delve into the process of solving inequalities, with a particular focus on identifying the property that justifies the transition between specific steps. We will use a concrete example to illustrate the process and highlight the importance of understanding the underlying principles of inequality manipulation.

Problem Statement

Consider the following inequality problem, where Michael has attempted to solve it in a step-by-step manner:

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

The question we aim to answer is: What property justifies the work shown between Step 3 and Step 4?

Breaking Down the Steps

To determine the property used, let's carefully examine the transition from Step 3 to Step 4.

Step 3: $-6x - 8 < -2$ Step 4: $-6x < 6$

Observing the change, we see that the term '-8' has been eliminated from the left side of the inequality. To achieve this, we must have added '8' to both sides of the inequality.

The Addition Property of Inequality

The property that allows us to add the same value to both sides of an inequality without changing its validity is known as the Addition Property of Inequality. This property is a fundamental concept in algebra and is crucial for solving inequalities.

Addition Property of Inequality: If a < b, then a + c < b + c for any real number c.

In our case, we added '8' to both sides of the inequality in Step 3:

$-6x - 8 + 8 < -2 + 8$

This simplifies to:

$-6x < 6$

Which is exactly Step 4.

Therefore, the property that justifies the work shown between Step 3 and Step 4 is the Addition Property of Inequality.

Why Other Options Are Incorrect

Let's briefly discuss why the other options provided in a multiple-choice setting would be incorrect:

Transitive Property: The transitive property typically applies when comparing three or more quantities. For example, if a < b and b < c, then a < c. This property is not directly used in the transition from Step 3 to Step 4.

Deep Dive into Solving Inequalities

To further enhance your understanding, let's delve deeper into the process of solving inequalities. Solving inequalities is similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number reverses the inequality sign. Understanding the properties that govern inequalities is paramount for obtaining correct solutions.

Key Properties of Inequalities

Besides the Addition Property, several other key properties are essential for manipulating inequalities:

Subtraction Property of Inequality: If a < b, then a - c < b - c for any real number c. This property is analogous to the Addition Property and allows us to subtract the same value from both sides of an inequality.
Multiplication Property of Inequality:
- If a < b and c > 0, then ac < bc (multiplying by a positive number preserves the inequality).
- If a < b and c < 0, then ac > bc (multiplying by a negative number reverses the inequality).
Division Property of Inequality:
- If a < b and c > 0, then a/c < b/c (dividing by a positive number preserves the inequality).
- If a < b and c < 0, then a/c > b/c (dividing by a negative number reverses the inequality).

These properties are the foundation for solving various types of inequalities, including linear, compound, and absolute value inequalities.

Applying the Properties: A Detailed Example

Let's revisit the original problem and walk through each step, explicitly stating the property used:

Step 1: $-6(x+3)+10<-2$

This is the initial inequality.

Step 2: $-6x-18+10<-2$

Property Used: Distributive Property. We distributed the -6 across the terms inside the parentheses: -6(x + 3) = -6x - 18.

Step 3: $-6x-8<-2$

Property Used: Simplification. We combined the constant terms: -18 + 10 = -8.

Step 4: $-6x<6$

Property Used: Addition Property of Inequality. We added 8 to both sides: -6x - 8 + 8 < -2 + 8, which simplifies to -6x < 6.

Step 5: $x>-1$

Property Used: Division Property of Inequality. We divided both sides by -6. Since we divided by a negative number, we reversed the inequality sign: (-6x) / -6 > 6 / -6, which simplifies to x > -1.

By explicitly stating the property used in each step, we gain a deeper understanding of the solution process and ensure the correctness of our work. This detailed approach is crucial for tackling more complex inequalities.

Common Mistakes to Avoid

When solving inequalities, it's important to be aware of common pitfalls that can lead to incorrect solutions. Here are some key mistakes to avoid:

Forgetting to Reverse the Inequality Sign: The most frequent mistake is failing to reverse the inequality sign when multiplying or dividing both sides by a negative number. Remember, this step is crucial for maintaining the validity of the inequality.
Incorrectly Distributing Negative Signs: When distributing a negative number, ensure that the negative sign is applied to all terms inside the parentheses. A common error is to only apply the negative sign to the first term.
Combining Unlike Terms: Only combine like terms (terms with the same variable and exponent) when simplifying inequalities. For instance, you cannot combine -6x and -8 in the expression -6x - 8.
Misinterpreting the Solution Set: The solution to an inequality is often a range of values, not a single value. Represent the solution set correctly using interval notation or a number line.

By being mindful of these common mistakes, you can significantly improve your accuracy in solving inequalities.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used extensively in various fields, including:

Economics: Inequalities are used to model supply and demand, budget constraints, and profit maximization.
Engineering: Engineers use inequalities to design structures that can withstand certain loads and to ensure that systems operate within specified limits.
Computer Science: Inequalities are used in algorithm analysis, optimization problems, and resource allocation.
Statistics: Inequalities are used in hypothesis testing and confidence interval estimation.

For example, consider a scenario where a company wants to determine the number of units it needs to sell to make a profit. The company's profit can be expressed as an inequality, where the revenue (selling price per unit times the number of units sold) must be greater than the total cost (fixed costs plus variable costs per unit times the number of units sold). By solving this inequality, the company can determine the minimum number of units it needs to sell to achieve profitability.

Understanding the applications of inequalities helps to appreciate their practical significance and motivates the learning process.

Conclusion

In conclusion, the property that justifies the work between Step 3 and Step 4 in the given inequality problem is the Addition Property of Inequality. This property allows us to add the same value to both sides of an inequality without changing its validity. By understanding the fundamental properties of inequalities and practicing problem-solving, you can confidently tackle a wide range of inequality problems. Remember to pay close attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. Inequalities are a fundamental tool in mathematics and have widespread applications in various fields, making their mastery essential for success in STEM disciplines and beyond.

By meticulously walking through each step and explicitly stating the property used, we not only arrive at the correct answer but also reinforce the underlying principles of inequality manipulation. This approach is crucial for building a strong foundation in algebra and problem-solving skills. Remember to practice regularly and apply these concepts to diverse problems to solidify your understanding. The journey of mastering inequalities is not just about finding solutions; it's about developing a logical and analytical mindset that will serve you well in various aspects of life.

Understanding the Addition Property of Inequality is crucial for manipulating and solving inequalities effectively. This property, along with others like the Subtraction, Multiplication, and Division Properties, forms the bedrock of inequality manipulation. By mastering these properties, students can confidently navigate a wide range of inequality problems and apply these skills in various mathematical and real-world contexts. The ability to solve inequalities is not just a mathematical skill; it's a powerful tool for decision-making, problem-solving, and critical thinking in diverse fields such as economics, engineering, and computer science.

The ability to solve inequalities is a fundamental skill in mathematics with applications spanning various fields. Understanding the properties that govern inequalities, such as the Addition Property, is crucial for achieving accurate solutions. By breaking down the problem into steps and meticulously applying the relevant properties, students can develop a strong foundation in algebra and problem-solving. Moreover, recognizing the real-world applications of inequalities, from economics to engineering, underscores their importance and motivates further learning.

Therefore, the answer is the Addition Property of Inequality, a cornerstone concept for anyone venturing into the world of algebra and mathematical problem-solving. Mastering this property unlocks the door to solving a wide array of inequality problems, enabling students to confidently tackle challenges in mathematics and beyond. The journey of learning mathematics is not just about memorizing formulas and procedures; it's about developing a deep understanding of the underlying principles and their applications, fostering a lifelong appreciation for the power and beauty of mathematics.