Which Number Line Accurately Represents The Solution To The Inequality (1/4)x + 1 < 5?

In the realm of mathematics, inequalities play a crucial role in describing relationships where quantities are not necessarily equal. Unlike equations that assert the equality of two expressions, inequalities express the relative order or comparison between them. These comparisons can be less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Understanding and solving inequalities is fundamental in various mathematical disciplines, including algebra, calculus, and real analysis. Furthermore, the ability to represent the solutions of inequalities on a number line provides a visual and intuitive way to grasp the range of values that satisfy a given inequality. This article delves into the intricacies of solving inequalities and the art of depicting their solutions on a number line, using the specific example of the inequality (1/4)x + 1 < 5 as a guiding illustration. We will explore the step-by-step process of isolating the variable, determining the solution set, and accurately representing this solution set on a number line.

Before diving into the specifics of solving (1/4)x + 1 < 5, it's essential to solidify our understanding of inequalities in general. An inequality is a mathematical statement that compares two expressions using inequality symbols. These symbols dictate the relationship between the expressions, indicating whether one is less than, greater than, less than or equal to, or greater than or equal to the other. Unlike equations, which have a finite number of solutions (or no solutions at all), inequalities often have an infinite set of solutions. This is because inequalities define a range of values that satisfy the given condition. For instance, the inequality x > 3 represents all numbers greater than 3, an infinite set that includes 3.0001, 4, 100, and so on. Grasping this concept of a solution set as a range of values is crucial for effectively solving and interpreting inequalities. Understanding how inequalities differ from equations is paramount. Equations aim to find specific values that make two expressions equal, whereas inequalities identify a range of values that satisfy a comparative relationship. This distinction fundamentally alters the nature of the solution and its representation. For instance, the equation x + 2 = 5 has a single solution, x = 3. Conversely, the inequality x + 2 < 5 has a solution set encompassing all values less than 3, highlighting the infinite nature of inequality solutions.

Now, let's focus on solving the inequality (1/4)x + 1 < 5. The objective is to isolate the variable 'x' on one side of the inequality. This process involves applying algebraic operations to both sides of the inequality while maintaining its balance. The fundamental principle here is that any operation performed on one side must also be performed on the other to preserve the inequality's validity. This is akin to solving equations, with a crucial exception: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This rule is paramount and must be diligently observed to ensure accurate solutions.

Step 1: Isolate the term with 'x'. To begin, we subtract 1 from both sides of the inequality: (1/4)x + 1 - 1 < 5 - 1. This simplifies to (1/4)x < 4. This step is analogous to isolating the variable term in an equation, where we aim to get the term containing 'x' by itself on one side of the expression. Subtracting 1 from both sides maintains the inequality's balance, ensuring that the solution set remains unchanged. This is a fundamental algebraic manipulation that paves the way for further simplification.

Step 2: Solve for 'x'. Next, to isolate 'x', we multiply both sides of the inequality by 4: 4 * (1/4)x < 4 * 4. This yields x < 16. This step completes the isolation of 'x', providing us with the solution set. Multiplying both sides by 4, a positive number, does not affect the direction of the inequality sign. The result, x < 16, signifies that any value of 'x' less than 16 will satisfy the original inequality. This is the solution set, and it represents an infinite range of values.

Solution: The solution to the inequality (1/4)x + 1 < 5 is x < 16. This means that any value of x less than 16 will make the inequality true. For instance, if x = 10, then (1/4)(10) + 1 = 2.5 + 1 = 3.5, which is indeed less than 5. Conversely, if x = 20, then (1/4)(20) + 1 = 5 + 1 = 6, which is not less than 5, confirming that values greater than or equal to 16 do not satisfy the inequality. This reinforces the understanding that the solution x < 16 defines a specific range of values that make the inequality a true statement.

Now that we have determined the solution x < 16, the next step is to visually represent this solution set on a number line. A number line is a powerful tool for illustrating the range of values that satisfy an inequality. It provides a clear and intuitive depiction of the solution set, making it easier to comprehend the infinite nature of the solution.

Step 1: Draw a number line. Begin by drawing a straight line and marking a scale on it. This scale should include the critical value (in this case, 16) and some values on either side to provide context. The number line extends infinitely in both directions, represented by arrows at each end. This symbolizes that numbers continue indefinitely in both the positive and negative directions.

Step 2: Mark the critical value. Locate 16 on the number line. Since the inequality is x < 16 (and not x ≤ 16), we use an open circle at 16. An open circle indicates that 16 is not included in the solution set. If the inequality were x ≤ 16, we would use a closed circle (or a filled-in circle) to signify that 16 is included in the solution. This distinction is crucial for accurately representing the solution set.

Step 3: Shade the appropriate region. Since the solution includes all values less than 16, we shade the region of the number line to the left of 16. This shaded region represents all the numbers that satisfy the inequality x < 16. The arrow extending to the left indicates that the solution set continues infinitely in the negative direction. This visual representation provides a comprehensive understanding of the solution set, highlighting the range of values that make the inequality true.

Visual Representation: The number line will have an open circle at 16 and the line shaded to the left, indicating all numbers less than 16 are solutions.

Visualizing solutions on a number line is not merely a graphical exercise; it's a crucial step in comprehending the nature of inequalities. It transforms an abstract algebraic solution into a tangible representation, making it easier to grasp the range of values that satisfy the inequality. This visualization is particularly beneficial for students learning about inequalities, as it provides a concrete way to understand the concept of a solution set as a range rather than a single value. Furthermore, representing solutions on a number line is invaluable when dealing with compound inequalities, which involve multiple inequalities combined with 'and' or 'or'. In these cases, the number line provides a clear visual representation of the intersection or union of the solution sets, simplifying the process of determining the overall solution. The ability to accurately depict solutions on a number line is a fundamental skill in mathematics, fostering a deeper understanding of inequalities and their applications.

When solving and representing inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls is crucial for maintaining accuracy and avoiding errors. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This rule is paramount, and neglecting it will invariably result in an incorrect solution set. For instance, if we have the inequality -2x < 4, dividing both sides by -2 requires flipping the inequality sign to obtain x > -2. Failing to do so would yield the incorrect solution x < -2.

Another common mistake occurs when interpreting the inequality symbol. It's essential to distinguish between strict inequalities (< and >) and inclusive inequalities (≤ and ≥). Strict inequalities use open circles on the number line to indicate that the endpoint is not included in the solution set, while inclusive inequalities use closed circles to signify that the endpoint is part of the solution. Misinterpreting these symbols can lead to inaccurate representations of the solution set. Additionally, errors can arise during the algebraic manipulation process, such as incorrectly distributing a negative sign or combining like terms. Careful attention to detail and a systematic approach are essential for minimizing these errors.

Finally, it's crucial to verify the solution by substituting values from the solution set back into the original inequality. This step helps confirm that the solution is correct and identifies any potential errors made during the solving process. By being mindful of these common mistakes and adopting a meticulous approach, one can confidently solve and represent inequalities accurately.

Solving inequalities and representing their solutions on a number line is a fundamental skill in mathematics. It provides a powerful way to describe and visualize relationships between quantities that are not necessarily equal. In this article, we explored the step-by-step process of solving the inequality (1/4)x + 1 < 5, which involved isolating the variable 'x' and determining the solution set x < 16. We then delved into the art of representing this solution on a number line, using an open circle at 16 and shading the region to the left to indicate all values less than 16. The number line serves as a visual aid, making it easier to comprehend the infinite nature of the solution set.

We also emphasized the importance of avoiding common mistakes, such as forgetting to reverse the inequality sign when multiplying or dividing by a negative number and misinterpreting inequality symbols. These errors can lead to inaccurate solutions, highlighting the need for careful attention to detail. By mastering the techniques outlined in this article, students and practitioners can confidently solve and represent inequalities, fostering a deeper understanding of mathematical relationships and problem-solving strategies. The ability to work with inequalities is not only crucial for academic success but also essential for real-world applications, where comparing quantities and determining ranges of values are commonplace tasks. Understanding inequalities and their graphical representations enhances mathematical literacy and empowers individuals to make informed decisions in various contexts.