Solving And Representing Inequalities The Case Of 3(8-4x) Less Than 6(x-5)

In mathematics, inequalities play a crucial role in defining ranges and sets of solutions. A number line is a powerful visual tool to represent these solutions. This article delves into solving the inequality $3(8-4x) < 6(x-5)$ and accurately representing its solution set on a number line. By understanding the step-by-step process, readers will gain a solid grasp of how to manipulate inequalities and interpret their solutions graphically. This is a fundamental skill in algebra and is essential for more advanced mathematical concepts. This article aims to provide a comprehensive guide, breaking down each step with clear explanations and examples, ensuring that the reader can confidently tackle similar problems in the future. Understanding inequalities and number lines is not just about solving equations; it's about developing a deeper understanding of mathematical relationships and problem-solving strategies. We will explore the core principles behind inequalities, the properties that govern their manipulation, and how these principles translate into a visual representation on a number line. This approach will empower readers to not only find the correct answer but also to understand the underlying mathematical concepts.

Breaking Down the Inequality

To accurately represent the solution set for the inequality $3(8-4x) < 6(x-5)$ on a number line, we must first systematically solve the inequality. This involves isolating the variable x on one side of the inequality sign. The process begins by applying the distributive property to both sides of the inequality. The distributive property states that for any numbers a, b, and c, $a(b+c) = ab + ac$. Applying this property to our inequality, we get: $3 * 8 - 3 * 4x < 6 * x - 6 * 5$, which simplifies to $24 - 12x < 6x - 30$. This step is crucial as it removes the parentheses and allows us to combine like terms. The distributive property is a cornerstone of algebraic manipulation, and mastering its application is essential for solving more complex equations and inequalities. It ensures that each term within the parentheses is correctly multiplied, maintaining the balance and integrity of the equation. Now that we've applied the distributive property, the next step is to consolidate the x terms and the constant terms. We aim to group all terms containing x on one side of the inequality and all constant terms on the other side. This is achieved by adding or subtracting terms from both sides of the inequality. Remember, the key principle is to maintain the balance of the inequality; any operation performed on one side must also be performed on the other side. This ensures that the inequality remains valid and the solution set remains unchanged. We will now delve into the specific steps for isolating x, ensuring each step is clearly explained and justified. Understanding the logic behind each operation is as important as the operation itself, as it builds a deeper understanding of the mathematical principles at play.

Isolating the Variable

Having simplified the inequality to $24 - 12x < 6x - 30$, our next goal is to isolate the variable x. To achieve this, we need to group the x terms on one side and the constant terms on the other. A common strategy is to move the x terms to the side that will result in a positive coefficient for x. In this case, adding $12x$ to both sides of the inequality will eliminate the $-12x$ term on the left side and result in a positive coefficient for x on the right side. This gives us: $24 - 12x + 12x < 6x - 30 + 12x$, which simplifies to $24 < 18x - 30$. This step is crucial because it simplifies the inequality and brings us closer to isolating x. Adding the same term to both sides of an inequality is a fundamental operation that preserves the inequality's validity. It's like adding the same weight to both sides of a balance scale – the scale remains balanced. Now that we have the x terms on one side, we need to move the constant terms to the other side. To do this, we add 30 to both sides of the inequality: $24 + 30 < 18x - 30 + 30$, which simplifies to $54 < 18x$. By adding 30 to both sides, we have successfully isolated the x term on the right side of the inequality. This step mirrors the previous one, reinforcing the principle of maintaining balance in an inequality. With the variable term and constant term separated, we are now in a position to solve for x directly. The next step involves dividing both sides of the inequality by the coefficient of x.

Solving for x

Now that we have the inequality in the form $54 < 18x$, the final step in solving for x is to divide both sides of the inequality by 18. This isolates x on the right side of the inequality, giving us the solution set. Dividing both sides by 18, we get: $54 / 18 < (18x) / 18$, which simplifies to $3 < x$. This can also be written as $x > 3$. This result tells us that the solution set includes all values of x that are greater than 3. It's crucial to remember that when dividing (or multiplying) both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. However, in this case, we are dividing by a positive number (18), so the direction of the inequality sign remains unchanged. The solution $x > 3$ represents an infinite set of numbers. Any number greater than 3 satisfies the original inequality. This includes numbers like 3.0001, 4, 5, 10, 100, and so on. Understanding that inequalities often have an infinite number of solutions is a key concept in algebra. Now that we have the solution set, the next step is to represent this set on a number line. This visual representation provides a clear and intuitive understanding of the solution and its range. The number line will show all the values of x that satisfy the inequality, and it will also highlight the boundary point (in this case, 3) and whether or not it is included in the solution set.

Representing the Solution on a Number Line

To represent the solution set $x > 3$ on a number line, we need to understand the conventions used for depicting inequalities graphically. A number line is a horizontal line that represents all real numbers. Numbers increase as you move from left to right, with zero at the center. To represent $x > 3$, we first locate the number 3 on the number line. Since the inequality is strictly greater than ( > ), the number 3 itself is not included in the solution set. This is represented by using an open circle (or parenthesis) at the point 3 on the number line. An open circle indicates that the endpoint is not part of the solution. If the inequality were greater than or equal to ($\\\geq$), we would use a closed circle (or bracket) to indicate that the endpoint is included. Next, we need to indicate all the numbers greater than 3. This is done by drawing a line (or ray) extending to the right from the open circle at 3. The line continues indefinitely to the right, often with an arrowhead, indicating that the solution set includes all numbers greater than 3, extending to positive infinity. The arrowhead is a crucial part of the representation, as it visually communicates the unbounded nature of the solution set. The combination of the open circle and the ray extending to the right provides a clear and concise visual representation of the solution $x > 3$. This visual representation is not just a way to display the solution; it also helps in understanding the concept of inequalities and solution sets more intuitively. By looking at the number line, one can easily identify which numbers satisfy the inequality and which do not. In conclusion, the number line representation of $x > 3$ will have an open circle at 3 and a ray extending to the right, visually capturing all numbers greater than 3.

Conclusion

In summary, we have successfully solved the inequality $3(8-4x) < 6(x-5)$ and determined its solution set to be $x > 3$. We then learned how to accurately represent this solution set on a number line, using an open circle at 3 to indicate that 3 is not included in the solution and a ray extending to the right to represent all numbers greater than 3. This process involves a series of algebraic manipulations, including the distributive property, combining like terms, and isolating the variable. Each step is crucial to arriving at the correct solution. Furthermore, understanding the conventions for representing inequalities on a number line is essential for visualizing and interpreting the solution set. Number lines provide a powerful visual aid for understanding inequalities and their solutions. They allow us to see the range of values that satisfy the inequality and to easily identify the boundary points. This skill is fundamental to understanding more complex mathematical concepts. This exercise demonstrates the interconnectedness of algebra and graphical representation in mathematics. Being able to solve an inequality and then represent its solution on a number line is a testament to a solid understanding of these fundamental concepts. Mastering these skills will provide a strong foundation for further exploration in mathematics. Therefore, the ability to solve inequalities and represent their solutions on number lines is a valuable skill that is applicable in various mathematical contexts.