Inequality Properties Understanding The Steps And Justifications

In the realm of mathematics, inequalities play a crucial role in describing relationships where quantities are not necessarily equal. Solving inequalities involves a series of steps, each justified by specific mathematical properties. Understanding these properties is essential for accurately manipulating inequalities and arriving at the correct solution. This article delves into the process of solving a particular inequality, highlighting the properties that underpin each step. We will dissect the solution provided for the inequality $-6(x+3)+10<-2$, identify the property that justifies the transition from one step to the next, and provide a comprehensive understanding of the underlying mathematical principles. Let's embark on this journey of mathematical exploration to enhance our problem-solving skills and deepen our grasp of inequality manipulation.

Dissecting the Inequality Solution

Let's examine the steps taken to solve the inequality:

• 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 core question we aim to address is: What property justifies the work shown between Step 3 and Step 4? To answer this, we must meticulously analyze the transformation that occurs between these two steps.

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

The initial step presents the inequality in its original form. This sets the stage for the subsequent simplification process. The expression on the left-hand side involves distribution and constant terms, which need to be addressed to isolate the variable x. Understanding the order of operations and the properties of real numbers is crucial for navigating this initial step successfully. The goal is to transform the inequality into a more manageable form while preserving its fundamental meaning.

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

The transition from Step 1 to Step 2 involves the application of the distributive property. This property states that a(b + c) = ab + ac. In this case, -6 is distributed across the terms inside the parentheses (x + 3), resulting in -6x - 63, which simplifies to -6x - 18. This step is crucial for removing the parentheses and bringing the inequality closer to a form where the variable can be isolated. The distributive property is a cornerstone of algebraic manipulation and is frequently used in solving equations and inequalities.

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

Step 3 is a simplification of Step 2. Here, the constant terms on the left-hand side, -18 and +10, are combined. -18 + 10 equals -8. This step streamlines the inequality, making it easier to isolate the term containing the variable x. Combining like terms is a fundamental algebraic technique that helps to simplify expressions and equations, making them more manageable to solve. This step demonstrates the importance of arithmetic operations in the context of algebraic manipulation.

Step 4: $-6x<6$

Now, let's focus on the critical transition from Step 3 ($-6x-8<-2$) to Step 4 ($-6x<6$). To isolate the term with x, we need to eliminate the constant term -8 on the left-hand side. This is achieved by adding 8 to both sides of the inequality. Adding the same value to both sides maintains the balance of the inequality. This is a direct application of the Addition Property of Inequality. The Addition Property of Inequality states that if a < b, then a + c < b + c for any real number c. This property is crucial for manipulating inequalities while preserving the relationship between the two sides.

Step 5: $x>-1$

Step 5 is obtained from Step 4 by dividing both sides of the inequality by -6. However, there's a crucial detail to remember when dividing (or multiplying) an inequality by a negative number: you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the quantities involved, which affects their relative order. So, when we divide -6x < 6 by -6, we get x > -1. The sign flips from "less than" to "greater than". This step highlights the Division Property of Inequality, which states that if a < b and c is a negative number, then a/ c > b/ c. This property is a critical concept in solving inequalities and must be applied carefully to avoid errors.

Identifying the Justifying Property

Therefore, the property that justifies the work shown between Step 3 and Step 4 is the Addition Property of Inequality. We added 8 to both sides of the inequality to isolate the term containing x. This property ensures that the inequality remains balanced and the solution set is preserved.

Why Other Options are Incorrect

To further solidify our understanding, let's briefly discuss why the other answer choices might be incorrect:

• Transitive Property: The transitive property typically applies to statements of equality or order where if a = b and b = c, then a = c. It doesn't directly justify adding a constant to both sides of an inequality.

Deep Dive into the Properties of Inequalities

To master the art of solving inequalities, it's essential to have a firm grasp of the properties that govern their manipulation. These properties provide the foundation for accurately isolating variables and determining the solution set. Let's explore some of the key properties in detail:

1. Addition Property of Inequality: Maintaining Balance

The Addition Property of Inequality is a fundamental principle that allows us to add the same value to both sides of an inequality without altering the solution set. This property is crucial for isolating variables and simplifying inequalities. Mathematically, it can be expressed as follows:

If a < b, then a + c < b + c for any real number c.

Similarly, if a > b, then a + c > b + c for any real number c. This property holds true regardless of whether the value being added (c) is positive, negative, or zero. The key is that the same value is added to both sides, preserving the relative order of the quantities.

To illustrate, consider the inequality x - 5 < 3. To isolate x, we can add 5 to both sides: (x - 5) + 5 < 3 + 5, which simplifies to x < 8. The Addition Property of Inequality guarantees that the solution set remains unchanged throughout this process.

2. Subtraction Property of Inequality: A Reverse Operation

The Subtraction Property of Inequality is closely related to the Addition Property and serves as its inverse operation. It states that subtracting the same value from both sides of an inequality does not change the solution set. The mathematical representation is:

If a < b, then a - c < b - c for any real number c.

Likewise, if a > b, then a - c > b - c for any real number c. Just like the Addition Property, the Subtraction Property holds true for any real number c, whether positive, negative, or zero.

For example, consider the inequality x + 2 > 7. To isolate x, we can subtract 2 from both sides: (x + 2) - 2 > 7 - 2, which simplifies to x > 5. The Subtraction Property ensures that the inequality remains balanced and the solution set is preserved.

3. Multiplication Property of Inequality: A Critical Consideration

The Multiplication Property of Inequality introduces a crucial consideration: the sign of the number being multiplied. Multiplying both sides of an inequality by a positive number preserves the direction of the inequality, while multiplying by a negative number reverses the direction. This distinction is critical for accurate inequality manipulation.

• Multiplying by a Positive Number:

  If a < b and c > 0, then a c < b c.

  Similarly, if a > b and c > 0, then a c > b c.

  In this case, the inequality sign remains unchanged.

• Multiplying by a Negative Number:

  If a < b and c < 0, then a c > b c.

  Likewise, if a > b and c < 0, then a c < b c.

  Here, the inequality sign is reversed.

Consider the inequality x/2 < 4. To isolate x, we can multiply both sides by 2 (a positive number): (x/2) * 2 < 4 * 2, which simplifies to x < 8. The inequality sign remains unchanged.

Now, consider the inequality -3x < 9. To isolate x, we need to divide both sides by -3 (a negative number). This will reverse the inequality sign: (-3x) / -3 > 9 / -3, which simplifies to x > -3. The sign flips from "less than" to "greater than".

4. Division Property of Inequality: Mirroring Multiplication

The Division Property of Inequality mirrors the Multiplication Property in its behavior. Dividing both sides of an inequality by a positive number preserves the direction of the inequality, while dividing by a negative number reverses the direction.

• Dividing by a Positive Number:

  If a < b and c > 0, then a/ c < b/ c.

  Similarly, if a > b and c > 0, then a/ c > b/ c.

  The inequality sign remains unchanged.

• Dividing by a Negative Number:

  If a < b and c < 0, then a/ c > b/ c.

  Likewise, if a > b and c < 0, then a/ c < b/ c.

  The inequality sign is reversed.

These properties are the bedrock of solving inequalities. A thorough understanding of these properties, particularly the Multiplication and Division Properties, is critical for accurate manipulation and arriving at the correct solution set.

Solving Inequalities: A Comprehensive Approach

Solving inequalities requires a systematic approach, combining algebraic manipulation with a solid understanding of the properties discussed above. Here's a step-by-step guide to effectively tackle inequalities:

1. Simplify Both Sides: Begin by simplifying both sides of the inequality. This may involve distributing, combining like terms, or clearing fractions or decimals. The goal is to reduce the inequality to its simplest form, making it easier to isolate the variable.

2. Isolate the Variable Term: Use the Addition and Subtraction Properties of Inequality to move all terms containing the variable to one side of the inequality and all constant terms to the other side. This process aims to isolate the variable term, bringing you closer to the solution.

3. Isolate the Variable: Apply the Multiplication or Division Property of Inequality to isolate the variable completely. Remember the crucial rule: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. This step unveils the solution set for the inequality.

4. Express the Solution: Express the solution in the appropriate format. This may involve using inequality notation (e.g., x > 3), interval notation (e.g., (3, ∞)), or graphing the solution on a number line. The choice of representation depends on the context of the problem and the desired level of detail.

5. Check the Solution: It's always a good practice to check your solution by substituting a value from the solution set back into the original inequality. This helps to ensure that your solution is correct and that you haven't made any errors in the manipulation process.

Real-World Applications of Inequalities

Inequalities aren't confined to the realm of textbooks and classrooms; they have numerous real-world applications across various fields. They are used to model constraints, optimize resources, and make informed decisions. Here are a few examples:

• Budgeting and Finance: Inequalities are used to represent budget constraints, such as the maximum amount of money that can be spent on a particular item or service. They help individuals and businesses make financial decisions within their limitations.
• Optimization Problems: In optimization problems, inequalities are used to define constraints on the variables involved. For instance, a company might use inequalities to represent production capacity, resource availability, or demand requirements. The goal is to find the optimal solution that satisfies all constraints.
• Scientific Modeling: Inequalities are used to model physical phenomena, such as the range of temperatures within which a chemical reaction can occur or the minimum amount of force required to move an object. They help scientists understand and predict the behavior of complex systems.
• Computer Science: Inequalities are used in algorithms and data structures, such as sorting algorithms and search algorithms. They help to define the conditions under which certain operations should be performed.

By mastering the art of solving inequalities, you equip yourself with a powerful tool for tackling real-world problems and making informed decisions in various domains.

Conclusion: Mastering Inequalities

In conclusion, solving inequalities involves a systematic application of algebraic principles and a keen understanding of the properties that govern their manipulation. The Addition Property of Inequality, which justifies adding the same value to both sides of an inequality, played a pivotal role in the step-by-step solution we analyzed. By mastering these properties and practicing problem-solving techniques, you can confidently navigate the world of inequalities and unlock their potential in various mathematical and real-world contexts. Remember to always pay close attention to the sign of the number you are multiplying or dividing by, as this will determine whether you need to reverse the direction of the inequality sign. With consistent practice and a solid grasp of the underlying principles, you can become proficient in solving inequalities and applying them to a wide range of problems.

What property justifies the transition from Step 3 to Step 4 in the provided inequality solution?

Inequality Properties Understanding the Steps and Justifications