Solving Inequalities Representing 3(8-4x) < 6(x-5) On A Number Line

In the realm of mathematics, inequalities serve as powerful tools for expressing relationships between quantities that are not necessarily equal. They allow us to define a range of possible values that satisfy a given condition, providing a more flexible and nuanced way to represent mathematical relationships than simple equations. Among the various methods for visualizing and understanding solutions to inequalities, the number line stands out as a particularly intuitive and effective approach. In this comprehensive exploration, we will delve into the process of identifying the number line that accurately represents the solution set for the inequality 3(8-4x) < 6(x-5). This exploration will not only enhance your understanding of solving inequalities but also equip you with the skills to interpret and represent solutions graphically on a number line. Let's begin by carefully dissecting the inequality and gradually unraveling its solution set, paving the way for a clear and concise graphical representation.

The solution set of an inequality encompasses all the values that, when substituted for the variable, make the inequality a true statement. Unlike equations, which typically have a finite number of solutions, inequalities often have an infinite number of solutions. This is because the solution set represents a range of values rather than specific points. For instance, the inequality x > 2 includes all numbers greater than 2, which is an infinite set. Understanding this concept is crucial for grasping the nature of inequality solutions and how they are represented on a number line. As we progress, we will see how this infinite nature manifests itself in the graphical representation of the solution set.

The number line is a visual representation of all real numbers, extending infinitely in both positive and negative directions. It serves as a powerful tool for visualizing mathematical concepts, including inequalities. Each point on the number line corresponds to a unique real number, and intervals on the number line represent ranges of values. When representing the solution set of an inequality on a number line, we use specific conventions to indicate whether the endpoints of the interval are included or excluded. Open circles (or parentheses) are used to denote endpoints that are not included in the solution set, while closed circles (or brackets) are used to denote endpoints that are included. This distinction is critical for accurately representing the solution set of an inequality. In the following sections, we will apply these conventions to the specific inequality at hand, constructing a number line that precisely captures its solution set.

To effectively decode which number line represents the solution set for the inequality 3(8-4x) < 6(x-5), it is paramount to first grasp the fundamental concepts of inequalities and solution sets. An inequality, unlike an equation, expresses a relationship where two values are not necessarily equal. Instead of an equals sign (=), inequalities employ symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols establish a range of values that satisfy the given condition, providing a broader scope than the specific solutions offered by equations.

The solution set of an inequality is the collection of all values that, when substituted for the variable, render the inequality a true statement. This set can encompass a finite number of solutions, as seen in simple cases, or, more commonly, an infinite range of values. The inequality x > 2, for example, has an infinite solution set, including all numbers greater than 2. Conversely, an equation like x = 2 has only one solution. This distinction is vital for understanding why inequalities are often represented using intervals on a number line, as we will see shortly.

The number line serves as a visual aid in representing the solution sets of inequalities. It is a straight line where each point corresponds to a real number, extending infinitely in both positive and negative directions. This visual tool allows us to map the range of values that satisfy an inequality, providing a clear picture of the solution set. When representing an inequality on a number line, we use open and closed circles (or parentheses and brackets) to denote whether the endpoints of the interval are included or excluded. An open circle (or parenthesis) indicates that the endpoint is not part of the solution set, typically used for strict inequalities (< or >), while a closed circle (or bracket) signifies that the endpoint is included, used for inequalities with ≤ or ≥. The shading of the number line then indicates the range of values that satisfy the inequality, completing the visual representation of the solution set.

To determine the correct number line representation for the solution set of the inequality 3(8-4x) < 6(x-5), we must systematically solve the inequality. This process involves a series of algebraic manipulations to isolate the variable x and identify the range of values that satisfy the given condition. Each step must be carefully executed to maintain the integrity of the inequality and avoid errors that could lead to an incorrect solution set.

Our initial step is to simplify both sides of the inequality by applying the distributive property. This involves multiplying the constants outside the parentheses by each term inside the parentheses. On the left side, we multiply 3 by both 8 and -4x, resulting in 24 - 12x. On the right side, we multiply 6 by both x and -5, yielding 6x - 30. Thus, the inequality becomes 24 - 12x < 6x - 30. This simplification prepares the inequality for further manipulation, allowing us to combine like terms and isolate the variable.

Next, we aim to gather all terms containing x on one side of the inequality and all constant terms on the other side. To achieve this, we add 12x to both sides of the inequality, which eliminates the -12x term on the left side. This yields 24 < 18x - 30. Subsequently, we add 30 to both sides to isolate the x term further, resulting in 54 < 18x. These steps are crucial for isolating the variable and determining the range of values that satisfy the inequality.

Finally, to completely isolate x, we divide both sides of the inequality by 18. This gives us 3 < x, which is equivalent to x > 3. This final inequality reveals the solution set: all values of x that are greater than 3. It is essential to recognize that dividing by a positive number does not change the direction of the inequality sign. However, if we were to divide by a negative number, we would need to reverse the direction of the inequality sign. With the solution now clearly defined, we can proceed to represent it graphically on a number line.

Once we have algebraically solved the inequality 3(8-4x) < 6(x-5) and arrived at the solution x > 3, the next crucial step is to represent this solution set graphically on a number line. The number line provides a visual representation of all real numbers, allowing us to easily depict the range of values that satisfy the inequality. This graphical representation is invaluable for understanding the solution set and communicating it effectively.

To accurately represent the solution x > 3 on a number line, we first need to locate the critical value, which in this case is 3. This value serves as the boundary for the solution set. Since the inequality is strictly greater than (x > 3), the value 3 itself is not included in the solution set. To indicate this exclusion, we use an open circle (or a parenthesis) at the point corresponding to 3 on the number line. This convention signifies that the solution set includes all numbers greater than 3 but not 3 itself.

Next, we need to represent the range of values that satisfy the inequality. Since x > 3, the solution set includes all numbers to the right of 3 on the number line. To visually indicate this, we draw an arrow extending from the open circle at 3 towards the positive infinity direction. This arrow signifies that all numbers greater than 3 are part of the solution set. The combination of the open circle at 3 and the arrow extending to the right provides a clear and concise graphical representation of the solution set for the inequality x > 3. This representation allows for easy interpretation and understanding of the solution.

With a clear understanding of how to represent the solution set of an inequality on a number line, we can now turn our attention to identifying the correct number line for the inequality 3(8-4x) < 6(x-5). We have already determined that the solution to this inequality is x > 3. This means that the correct number line representation will have an open circle at 3, indicating that 3 is not included in the solution, and an arrow extending to the right, signifying that all values greater than 3 are part of the solution set.

When presented with multiple number line options, the key is to carefully examine each one and compare it to the characteristics of the solution set we have derived. Look for a number line that has an open circle at 3 and is shaded or has an arrow pointing to the right. Any number line that has a closed circle at 3, shading or an arrow pointing to the left, or any other variation does not accurately represent the solution x > 3 and can be eliminated.

The process of identifying the correct number line is a visual matching task. It requires a precise understanding of the conventions used in representing inequalities on a number line and the ability to translate the algebraic solution into a graphical representation. By carefully analyzing the number line options and comparing them to the solution x > 3, we can confidently identify the one that accurately depicts the solution set of the inequality.

When working with inequalities and their graphical representations on a number line, there are several common mistakes that students often make. Being aware of these potential pitfalls can help you avoid errors and ensure accurate solutions. One common mistake is forgetting to distribute properly when simplifying inequalities. For example, in the inequality 3(8-4x) < 6(x-5), it is crucial to multiply both terms inside the parentheses by the constant outside. Failure to do so can lead to an incorrect inequality and, consequently, an incorrect solution set.

Another frequent error occurs when dealing with negative numbers. When multiplying or dividing both sides of an inequality by a negative number, it is essential to remember to reverse the direction of the inequality sign. For instance, if you have -2x < 6, dividing both sides by -2 requires flipping the inequality sign to get x > -3. Forgetting this rule can result in an incorrect solution set and an inaccurate number line representation.

A third common mistake involves the representation of the solution set on a number line. Students sometimes confuse the use of open and closed circles (or parentheses and brackets). An open circle (or parenthesis) indicates that the endpoint is not included in the solution set, while a closed circle (or bracket) signifies that the endpoint is included. It is crucial to use the correct notation based on the inequality symbol ( <, >, ≤, ≥ ). Additionally, be mindful of the direction of the arrow or shading, ensuring it accurately reflects the range of values that satisfy the inequality.

In summary, deciphering the number line that represents the solution set for the inequality 3(8-4x) < 6(x-5) involves a systematic approach encompassing algebraic manipulation and graphical representation. We began by understanding the fundamental concepts of inequalities and solution sets, emphasizing the distinction between strict and inclusive inequalities. We then meticulously solved the inequality, arriving at the solution x > 3. This involved distributing, combining like terms, and isolating the variable, ensuring each step preserved the integrity of the inequality.

The graphical representation of the solution set on a number line is a critical step. We learned that an open circle at 3 signifies that 3 is not included in the solution, while the arrow extending to the right indicates that all values greater than 3 are part of the solution set. This visual representation provides a clear and concise depiction of the solution, making it easy to understand and communicate.

Identifying the correct number line requires a careful comparison of the graphical representation we derived with the given options. We looked for a number line with an open circle at 3 and an arrow pointing to the right, ensuring it accurately reflects the solution x > 3. Finally, we addressed common mistakes to avoid, such as improper distribution, neglecting to reverse the inequality sign when multiplying or dividing by a negative number, and misinterpreting the notation on the number line. By mastering these concepts and techniques, you can confidently solve inequalities and represent their solutions graphically, enhancing your mathematical understanding and problem-solving skills.