Solving The Inequality 1/4 + X < 5/6 A Step-by-Step Guide

In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries. Understanding how to solve inequalities is fundamental for various applications, ranging from basic algebra to more complex calculus problems. This article delves into a step-by-step solution of the inequality ${ \frac{1}{4} + x < \frac{5}{6} }$, providing a clear and comprehensive guide suitable for students and anyone looking to refresh their knowledge. We'll explore the underlying concepts, walk through the solution process, and highlight the importance of each step. Grasping these concepts will not only help in solving this specific problem but will also build a strong foundation for tackling more complex inequalities in the future. Let's embark on this mathematical journey to unravel the solution to this inequality.

Understanding Inequalities

Before diving into the solution, it's essential to understand what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which show that two expressions are equal, inequalities show a relationship where one expression is either larger, smaller, or within a certain range compared to another. For instance, the inequality ${ a < b }$ indicates that ${ a }$ is less than ${ b }$, while ${ a > b }$ indicates that ${ a }$ is greater than ${ b }$. The symbols ≤ and ≥ include the possibility of equality, meaning ${ a ≤ b }$ implies that ${ a }$ is either less than or equal to ${ b }$. Understanding these symbols is crucial for interpreting and solving inequalities accurately. In the context of our problem, ${ \frac{1}{4} + x < \frac{5}{6} }$, we are looking for all values of ${ x }$ that, when added to ${ \frac{1}{4} }$, result in a value less than ${ \frac{5}{6} }$. This requires us to isolate ${ x }$ and determine the range of values that satisfy this condition. The principles applied to solving inequalities are similar to those used for equations, but there are some critical differences, especially when multiplying or dividing by negative numbers, which we will address later in this discussion.

Step-by-Step Solution of the Inequality

To solve the inequality ${ \frac{1}{4} + x < \frac{5}{6} }$, we follow a series of algebraic steps to isolate the variable ${ x }$ on one side of the inequality. The primary goal is to manipulate the inequality while preserving its truth, meaning that the solutions remain valid throughout the process. The first step in this process involves eliminating the fraction ${ \frac{1}{4} }$ from the left side of the inequality. To do this, we subtract ${ \frac{1}{4} }$ from both sides of the inequality. This operation maintains the balance of the inequality and allows us to gradually isolate ${ x }$. Subtracting ${ \frac{1}{4} }$ from both sides, we get:

${ x < \frac{5}{6} - \frac{1}{4} }$

Now, we need to simplify the right side of the inequality by subtracting the two fractions. To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 6 and 4 is 12, so we convert both fractions to have this denominator. We rewrite ${ \frac{5}{6} }$ as ${ \frac{10}{12} }$ (by multiplying both the numerator and denominator by 2) and ${ \frac{1}{4} }$ as ${ \frac{3}{12} }$ (by multiplying both the numerator and denominator by 3). This gives us:

${ x < \frac{10}{12} - \frac{3}{12} }$

Next, we perform the subtraction on the right side:

${ x < \frac{10 - 3}{12} }$

${ x < \frac{7}{12} }$

This final inequality, ${ x < \frac{7}{12} }$, represents the solution to the original inequality. It states that all values of ${ x }$ that are less than ${ \frac{7}{12} }$ will satisfy the initial condition. This solution is a range of values rather than a single value, which is characteristic of inequalities. Understanding this step-by-step process is crucial for solving similar inequalities and more complex mathematical problems. The solution highlights the importance of maintaining the balance of the inequality throughout the algebraic manipulations and correctly performing fraction operations.

Verifying the Solution

After finding a solution to an inequality, it's crucial to verify its correctness. This involves checking whether the solution obtained indeed satisfies the original inequality. In our case, we found that ${ x < \frac{7}{12} }$. To verify this, we can choose a value of ${ x }$ that is less than ${ \frac{7}{12} }$ and substitute it back into the original inequality, ${ \frac{1}{4} + x < \frac{5}{6} }$. A good choice for ${ x }$ would be 0, as it is clearly less than ${ \frac{7}{12} }$. Substituting ${ x = 0 }$ into the original inequality, we get:

${ \frac{1}{4} + 0 < \frac{5}{6} }$

${ \frac{1}{4} < \frac{5}{6} }$

To compare these fractions, we again find a common denominator, which is 12. Converting both fractions, we have:

${ \frac{3}{12} < \frac{10}{12} }$

This statement is true, as ${ \frac{3}{12} }$ is indeed less than ${ \frac{10}{12} }$. This verification step provides confidence that our solution, ${ x < \frac{7}{12} }$, is correct. It's important to note that while one successful verification doesn't guarantee the solution is correct for all values, it does provide a strong indication. For a more rigorous check, one could test another value less than ${ \frac{7}{12} }$ or a value greater than ${ \frac{7}{12} }$ to confirm that it does not satisfy the original inequality. Verifying solutions is a fundamental practice in mathematics, ensuring accuracy and reinforcing understanding of the problem-solving process. It helps in identifying potential errors and solidifies the concept being applied.

Common Mistakes and How to Avoid Them

When solving inequalities, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and understanding how to avoid them is essential for mastering the process. One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. For example, if you have the inequality ${ -2x < 4 }$, dividing both sides by -2 requires flipping the inequality sign, resulting in ${ x > -2 }$, not ${ x < -2 }$. This is because multiplying or dividing by a negative number reverses the order of the number line. Another common mistake is mishandling fractions, particularly when finding a common denominator or performing addition and subtraction. Ensuring fractions have a common denominator before performing these operations is crucial for accuracy. Carelessness in arithmetic can also lead to errors, such as incorrect subtraction or addition, which can significantly alter the solution. It's always a good practice to double-check each step, especially when dealing with multiple operations. Another pitfall is failing to distribute a negative sign correctly when it precedes a parenthesis. For instance, in the expression ${ -(x + 3) }$, the negative sign must be distributed to both ${ x }$ and 3, resulting in ${ -x - 3 }$. Neglecting to do this can lead to incorrect simplification and ultimately, an incorrect solution. Lastly, it's essential to remember that the solution to an inequality is often a range of values, not a single value. Understanding this concept helps in correctly interpreting the solution and avoiding the mistake of providing a single number as the answer. By being mindful of these common mistakes and practicing careful, step-by-step problem-solving, you can improve your accuracy and confidence in solving inequalities. Regularly verifying your solutions is also a valuable habit to ensure correctness.

Real-World Applications of Inequalities

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding inequalities allows us to model and solve problems involving constraints, limits, and ranges. In economics, inequalities are used to define budget constraints, showing the limits of what a consumer can afford given their income and the prices of goods. They also help in analyzing supply and demand curves, where the equilibrium price and quantity are determined by the intersection of these curves, but inequalities can describe situations where supply exceeds demand or vice versa. In engineering, inequalities are used extensively in design and optimization problems. For example, engineers might use inequalities to ensure that a structure can withstand a certain load or that a system operates within specific temperature or pressure limits. In computer science, inequalities are used in algorithm analysis to determine the efficiency and complexity of algorithms. They help in setting bounds on the time and space resources required to run an algorithm. In everyday life, inequalities are used in various decision-making processes. For instance, when planning a trip, one might use inequalities to determine the maximum distance that can be traveled within a given budget or time frame. Similarly, inequalities can be used to calculate the minimum amount of exercise needed to achieve a certain health goal. In the field of statistics, inequalities are used to define confidence intervals and hypothesis testing. They help in determining the range of values within which a population parameter is likely to fall, based on sample data. Moreover, inequalities play a crucial role in optimization problems, where the goal is to find the best solution among a set of possible solutions. This is widely used in operations research, logistics, and resource allocation. The ability to translate real-world scenarios into mathematical inequalities and solve them provides a powerful tool for analysis and decision-making. This highlights the practical significance of mastering inequalities and their applications.

Conclusion

In conclusion, solving the inequality ${ \frac{1}{4} + x < \frac{5}{6} }$ is a fundamental exercise in algebra that demonstrates the principles of manipulating inequalities. By subtracting ${ \frac{1}{4} }$ from both sides and simplifying the resulting expression, we determined that the solution is ${ x < \frac{7}{12} }$. This means that any value of ${ x }$ less than ${ \frac{7}{12} }$ will satisfy the original inequality. We also emphasized the importance of verifying the solution by substituting a value within the solution range back into the original inequality, ensuring accuracy. Throughout this discussion, we highlighted common mistakes to avoid, such as forgetting to flip the inequality sign when multiplying or dividing by a negative number and mishandling fractions. Understanding these pitfalls is crucial for accurate problem-solving. Furthermore, we explored the wide range of real-world applications of inequalities, demonstrating their practical significance in fields like economics, engineering, computer science, and everyday decision-making. Inequalities are not just abstract mathematical concepts; they are powerful tools for modeling and solving problems involving constraints, limits, and ranges. Mastering the techniques for solving inequalities is therefore an essential skill for students and professionals alike. By understanding the underlying principles and practicing regularly, one can develop confidence and proficiency in solving a wide variety of inequality problems. This knowledge not only enhances mathematical abilities but also provides a valuable framework for analytical thinking and problem-solving in diverse contexts. The journey through this inequality problem underscores the beauty and utility of mathematics in understanding and shaping the world around us.