Solving Inequalities Step By Step Guide For 3x > 5x + 8

Introduction

In this comprehensive guide, we will delve into the process of solving the inequality 3x > 5x + 8. Inequalities, a fundamental concept in mathematics, play a crucial role in various fields, including algebra, calculus, and real-world problem-solving. Understanding how to solve inequalities is essential for building a strong foundation in mathematics. This article provides a detailed, step-by-step explanation of the process, ensuring clarity and comprehension for learners of all levels. We will explore the underlying principles, demonstrate the algebraic manipulations involved, and interpret the solution set. By the end of this guide, you will be equipped with the knowledge and skills necessary to confidently tackle similar inequality problems.

Understanding Inequalities

Before we dive into the solution, let's first understand the basics of inequalities. Unlike equations, which represent equality between two expressions, inequalities represent a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols used to denote these relationships are >, <, ≥, and ≤, respectively. Inequalities are used to describe ranges of values rather than specific points. When solving inequalities, our goal is to isolate the variable on one side of the inequality sign, just as we do with equations. However, there's a crucial difference: when multiplying or dividing both sides of an inequality by a negative number, we must flip the inequality sign. This rule is essential to maintain the truth of the inequality. To further illustrate this point, consider the simple inequality 2 < 4. If we multiply both sides by -1 without flipping the sign, we get -2 < -4, which is false. However, if we flip the sign, we get -2 > -4, which is true. This seemingly small detail is critical for accurately solving inequalities. Beyond the basic symbols and the sign-flipping rule, understanding the properties of inequalities is crucial. The addition and subtraction properties state that adding or subtracting the same value from both sides of an inequality does not change the inequality. The multiplication and division properties dictate that multiplying or dividing both sides by a positive number does not change the inequality, but multiplying or dividing by a negative number requires flipping the sign. These properties are the foundation upon which we build our approach to solving inequalities. To grasp the significance of these properties, let's consider a real-world example. Suppose you have a budget for groceries and want to ensure that your total spending remains within that budget. This situation can be modeled using an inequality, where the total cost of your items must be less than or equal to your budget. Each property of inequalities helps you manipulate this inequality to determine how much you can spend on each item while staying within your budget. This practical application highlights the relevance and importance of understanding inequalities in everyday life.

Step-by-Step Solution of 3x > 5x + 8

Now, let's solve the inequality 3x > 5x + 8 step by step. Our first goal is to isolate the terms containing 'x' on one side of the inequality. This involves using the addition or subtraction property of inequalities. To do this, we can subtract 5x from both sides of the inequality. This operation maintains the balance of the inequality while moving the x-term from the right side to the left side. The result of this step is:

3x - 5x > 5x + 8 - 5x

Simplifying both sides, we get:

-2x > 8

Now, we need to isolate 'x' completely. We achieve this by dividing both sides of the inequality by the coefficient of 'x', which is -2. Remember the crucial rule: when dividing by a negative number, we must flip the inequality sign. This is a critical step in solving inequalities and is often a point where errors occur. Flipping the sign ensures that the direction of the inequality remains correct after the division. Performing the division, we have:

(-2x) / -2 < 8 / -2

Notice that the '>' sign has been flipped to '<'. Simplifying, we obtain:

x < -4

This is our solution. It states that 'x' is less than -4. This means any value of 'x' that is less than -4 will satisfy the original inequality. To fully understand this solution, it's beneficial to consider what this means graphically. The solution represents a range of values on the number line. Specifically, it represents all numbers to the left of -4, excluding -4 itself. We can visualize this on a number line by drawing an open circle at -4 and shading the region to the left. The open circle indicates that -4 is not included in the solution set. This graphical representation provides a visual confirmation of the solution and helps to solidify understanding. To further solidify understanding, let's consider a couple of examples. If we substitute x = -5 (which is less than -4) into the original inequality, we get 3(-5) > 5(-5) + 8, which simplifies to -15 > -25 + 8, or -15 > -17. This is a true statement, confirming that -5 is indeed a solution. On the other hand, if we substitute x = -3 (which is greater than -4), we get 3(-3) > 5(-3) + 8, which simplifies to -9 > -15 + 8, or -9 > -7. This is a false statement, confirming that -3 is not a solution. These examples demonstrate the validity of our solution and highlight the importance of understanding the direction of the inequality.

Verifying the Solution

To ensure the accuracy of our solution x < -4, we can perform a verification step. This involves substituting a value less than -4 into the original inequality and checking if the inequality holds true. Let's choose x = -5, which is clearly less than -4. Substituting this value into the original inequality, we get:

3(-5) > 5(-5) + 8

Simplifying both sides:

-15 > -25 + 8

-15 > -17

This statement is true, which confirms that our solution is correct. Choosing a value greater than -4, such as x = 0, should result in a false statement. Let's substitute x = 0 into the original inequality:

3(0) > 5(0) + 8

0 > 8

This statement is false, as expected. This further validates our solution x < -4. The process of verification is crucial in mathematics, particularly when solving equations and inequalities. It provides a means of checking for errors and ensuring that the obtained solution is correct. By substituting the solution back into the original problem, we can confirm whether the solution satisfies the given conditions. In the context of inequalities, verification becomes even more important because the solution often represents a range of values rather than a single value. Substituting a value within the solution range should result in a true statement, while substituting a value outside the range should result in a false statement. This process not only confirms the accuracy of the solution but also enhances understanding of the inequality and its implications. Furthermore, the verification step can help identify common errors, such as forgetting to flip the inequality sign when multiplying or dividing by a negative number. By systematically checking the solution, we can catch these errors and correct them, leading to a more accurate and reliable answer. In summary, the verification step is an integral part of the problem-solving process in mathematics. It provides a valuable tool for ensuring the accuracy of solutions and deepening understanding of the underlying concepts.

Representing the Solution on a Number Line

Visualizing the solution on a number line provides a clear and intuitive understanding of the solution set. Our solution, x < -4, represents all real numbers less than -4. To represent this on a number line, we draw a horizontal line and mark the point -4. Since the inequality is strictly less than (not less than or equal to), we use an open circle at -4 to indicate that -4 is not included in the solution set. Then, we shade the region to the left of -4, representing all numbers less than -4. This shaded region extends infinitely to the left, indicating that any number in this region satisfies the inequality. The number line representation offers a visual confirmation of the solution and helps to distinguish between strict inequalities (using open circles) and inclusive inequalities (using closed circles). For example, if the solution were x ≤ -4, we would use a closed circle at -4, indicating that -4 is included in the solution set. Similarly, if the solution were x > -4, we would use an open circle at -4 and shade the region to the right, representing all numbers greater than -4. The number line representation is a valuable tool for understanding and communicating solutions to inequalities. It provides a visual context that complements the algebraic solution and makes the solution set more tangible. Furthermore, it facilitates comparison between different inequalities and their solutions. For instance, we can easily compare the solutions x < -4 and x > -4 on a number line, visually highlighting the difference between the two solution sets. In addition to single inequalities, the number line can also be used to represent the solutions to compound inequalities, which involve two or more inequalities combined with logical operators such as