Solving Inequalities A Step-by-Step Guide To Find The Value Of X

In the realm of mathematics, inequalities play a crucial role in describing relationships between values that are not necessarily equal. They provide a powerful tool for expressing ranges and constraints, allowing us to solve a wide variety of problems. One common task involves finding the value(s) of a variable that satisfy a given inequality. In this comprehensive guide, we will delve into the process of solving inequalities, focusing on the specific example: $\frac{x}{3}-\frac{1}{3} < \frac{2x}{5} + \frac{1}{6}$. This step-by-step approach will equip you with the skills to tackle various inequality problems with confidence. Understanding inequalities is fundamental for various mathematical concepts, from linear programming to calculus, and is a vital skill for students and professionals alike. Solving inequalities involves manipulating expressions to isolate the variable, much like solving equations, but with an important distinction regarding the direction of the inequality when multiplying or dividing by a negative number. Let's embark on this journey to master the art of solving inequalities and uncover the value(s) of 'x' that make our given statement true. We will also explore the nuances of inequality symbols, the properties that govern their manipulation, and the graphical representation of solutions, ensuring a thorough understanding of the topic.

Step 1: Clearing Fractions to Simplify the Inequality

Our initial inequality, $\frac{x}{3}-\frac{1}{3} < \frac{2x}{5} + \frac{1}{6}$, contains fractions, which can make it cumbersome to work with directly. To simplify the inequality, our first step is to eliminate these fractions. We achieve this by finding the least common multiple (LCM) of the denominators, which are 3, 5, and 6. The LCM of these numbers is 30. Multiplying both sides of the inequality by 30 will clear the fractions, making the inequality easier to manipulate. This process is analogous to clearing fractions in equations, and it's a standard technique for solving inequalities involving rational expressions. Clearing fractions is a crucial step in simplifying inequalities as it transforms the inequality into a more manageable form, free from denominators. By multiplying both sides of the inequality by the LCM, we ensure that the denominators cancel out, resulting in an inequality involving only integers. This simplifies the subsequent algebraic manipulations, such as combining like terms and isolating the variable. The least common multiple (LCM) is the smallest multiple that is common to all the denominators. In this case, finding the LCM of 3, 5, and 6 helps us determine the appropriate number to multiply across the inequality to eliminate the fractions effectively. This foundational step sets the stage for solving the inequality accurately and efficiently.

$\qquad 30 \times (\frac{x}{3}-\frac{1}{3}) < 30 \times (\frac{2x}{5} + \frac{1}{6})$

$\qquad 10x - 10 < 12x + 5$

Step 2: Isolating the Variable Terms

Now that we have eliminated the fractions, our inequality is $10x - 10 < 12x + 5$. The next step in solving for x is to isolate the variable terms on one side of the inequality and the constant terms on the other side. To do this, we can subtract $10x$ from both sides of the inequality. This operation maintains the inequality because we are performing the same subtraction on both sides. This is a fundamental property of inequalities: adding or subtracting the same value from both sides does not change the direction of the inequality. Isolating the variable is a critical step in solving inequalities, as it brings all terms containing the variable to one side, allowing us to eventually determine the range of values that satisfy the inequality. By subtracting $10x$ from both sides, we effectively move the variable term from the left side to the right side, simplifying the inequality and bringing us closer to isolating x. This process mirrors the steps taken in solving linear equations, where the goal is to group like terms together to simplify the expression. The key here is to perform the same operation on both sides to maintain the balance and validity of the inequality.

$\qquad 10x - 10 - 10x < 12x + 5 - 10x$

$\qquad -10 < 2x + 5$

Step 3: Isolating the Constant Terms

Following the isolation of variable terms, we now have $-10 < 2x + 5$. Our next goal is to isolate the constant terms. To achieve this, we subtract 5 from both sides of the inequality. Similar to the previous step, this operation preserves the inequality since we are performing the same subtraction on both sides. Subtracting a constant from both sides is a valid algebraic manipulation that helps us isolate the variable term. Isolating constant terms is essential to further simplify the inequality and bring us closer to isolating x. By subtracting 5 from both sides, we move the constant term from the right side to the left side, making the inequality easier to solve. This step ensures that all constant terms are grouped together on one side, allowing us to consolidate them into a single constant value. Just like isolating variable terms, isolating constant terms is a standard technique in solving both equations and inequalities.

$\qquad -10 - 5 < 2x + 5 - 5$

$\qquad -15 < 2x$

Step 4: Solving for x

We've arrived at the inequality $-15 < 2x$. To finally solve for x, we need to divide both sides of the inequality by 2. Since we are dividing by a positive number, the direction of the inequality remains unchanged. Dividing by a positive number is a fundamental operation in solving inequalities, and it's important to remember that this operation does not flip the inequality sign. Solving for x involves performing the necessary operations to isolate the variable on one side of the inequality. By dividing both sides by 2, we isolate x and determine the range of values that satisfy the original inequality. This is the culmination of our algebraic manipulations, bringing us to the solution. The result provides the condition that x must satisfy, giving us the answer to the problem.

$\qquad \frac{-15}{2} < \frac{2x}{2}$

$\qquad -7.5 < x$

This can also be written as:

$\qquad x > -7.5$

Step 5: Interpreting the Solution

Our solution, $x > -7.5$, tells us that x can be any number greater than -7.5. In other words, all values of x that are larger than -7.5 will satisfy the original inequality. Understanding the solution in the context of the inequality is crucial for practical applications. The inequality $x > -7.5$ represents a range of values, not just a single value, and it's important to recognize this when interpreting the result. Interpreting the solution involves understanding the range of values that satisfy the inequality and how this relates to the original problem. The solution set includes all real numbers greater than -7.5, and this can be visualized on a number line, where an open circle at -7.5 indicates that -7.5 itself is not included in the solution. This interpretation allows us to make informed decisions and predictions based on the inequality.

Step 6: Identifying the Correct Option

Comparing our solution, $x > -7.5$, to the given options, we find that option (d) $x > -7 \frac{1}{2}$ is the correct answer. This step involves matching our calculated solution with the provided options, ensuring that we select the answer that accurately represents the range of values for x. Identifying the correct option is the final step in the problem-solving process, and it requires careful comparison and attention to detail. We must ensure that our solution aligns precisely with one of the options, confirming that we have correctly solved the inequality. This step validates our work and provides the definitive answer to the problem.

Additional Tips for Solving Inequalities

• Remember the sign flip: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
• Check your solution: After solving an inequality, it's always a good idea to check your solution by plugging in a value within the solution range into the original inequality. If the inequality holds true, your solution is likely correct.
• Graphing the solution: Visualizing the solution on a number line can help you understand the range of values that satisfy the inequality.

Conclusion

Solving inequalities is a fundamental skill in mathematics with broad applications. By following a systematic approach, such as the one outlined in this guide, you can confidently tackle a wide range of inequality problems. Remember to clear fractions, isolate variable and constant terms, and pay close attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. With practice, you'll master the art of solving inequalities and gain a deeper understanding of mathematical relationships.