Solving Linear Inequalities 7x + 2/6 - X < 5x + 4/3 - 4x A Step-by-Step Guide
Linear inequalities are a fundamental concept in mathematics, often encountered in algebra and calculus. Understanding how to solve them is crucial for various applications, from optimization problems to real-world scenarios. This article provides a detailed, step-by-step guide to solving the linear inequality 7x + 2/6 - x < 5x + 4/3 - 4x. We will break down each step, explaining the underlying principles and techniques involved. Whether you are a student learning about inequalities for the first time or someone looking to refresh your knowledge, this guide will help you master the process.
Understanding Linear Inequalities
Before diving into the solution, it’s important to understand what linear inequalities are and how they differ from linear equations. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which seek a specific value that makes the equation true, linear inequalities seek a range of values that satisfy the inequality. In our case, we are dealing with the inequality 7x + 2/6 - x < 5x + 4/3 - 4x, which we aim to solve for x. The solution will be a set of x values that make the inequality true.
The Importance of Solving Inequalities
Solving inequalities is not just an academic exercise; it has practical applications in various fields. In economics, inequalities are used to model supply and demand curves and to determine price ranges that ensure profitability. In engineering, inequalities help define safety margins and operational limits. In computer science, they are used in algorithm design and optimization. Understanding how to solve linear inequalities provides a valuable tool for problem-solving in these and many other areas.
Basic Principles for Solving Inequalities
Solving linear inequalities involves manipulating the inequality to isolate the variable on one side. The basic principles for solving inequalities are similar to those for solving equations, with one key difference: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. This is a crucial rule to remember, as it can affect the final solution. The other principles include adding or subtracting the same quantity from both sides, multiplying or dividing both sides by a positive number, and simplifying expressions by combining like terms. These principles allow us to systematically work towards isolating the variable and finding the solution set.
Step-by-Step Solution of 7x + 2/6 - x < 5x + 4/3 - 4x
Now, let’s tackle the given inequality step by step. We will break down the process into manageable parts, ensuring that each step is clear and easy to follow. By understanding each step, you’ll be able to apply the same techniques to solve other linear inequalities.
Step 1: Simplify Both Sides of the Inequality
The first step in solving any inequality is to simplify both sides by combining like terms. This makes the inequality easier to work with and helps prevent errors in later steps. In our case, we have the inequality 7x + 2/6 - x < 5x + 4/3 - 4x. On the left side, we can combine the terms 7x and -x, resulting in 6x. On the right side, we can combine the terms 5x and -4x, resulting in x. So, the simplified inequality becomes 6x + 2/6 < x + 4/3.
Simplifying both sides not only makes the inequality visually cleaner but also reduces the number of operations needed in subsequent steps. This is particularly important when dealing with more complex inequalities that involve multiple terms and operations. Taking the time to simplify at the beginning can save time and effort in the long run.
Step 2: Eliminate Fractions
Fractions can often complicate the process of solving inequalities. To eliminate them, we find the least common multiple (LCM) of the denominators and multiply both sides of the inequality by this LCM. In our inequality, 6x + 2/6 < x + 4/3, the denominators are 6 and 3. The LCM of 6 and 3 is 6. So, we multiply both sides of the inequality by 6.
Multiplying the left side by 6, we get 6 * (6x + 2/6) = 36x + 2. Multiplying the right side by 6, we get 6 * (x + 4/3) = 6x + 8. The inequality now becomes 36x + 2 < 6x + 8. Eliminating fractions makes the inequality easier to handle, as we are now dealing with whole numbers and integers, which are simpler to manipulate.
Step 3: Isolate the Variable Terms on One Side
To solve for x, we need to isolate the variable terms on one side of the inequality. This involves moving all terms containing x to one side and all constant terms to the other side. In our inequality, 36x + 2 < 6x + 8, we can subtract 6x from both sides to move the x term from the right side to the left side. This gives us 36x - 6x + 2 < 6x - 6x + 8, which simplifies to 30x + 2 < 8.
Isolating the variable terms is a crucial step in solving any inequality. It sets the stage for the final steps, where we will isolate the variable itself. By performing the same operation on both sides, we maintain the balance of the inequality and ensure that the solution remains valid.
Step 4: Isolate the Constant Terms on the Other Side
Now that we have isolated the variable terms on the left side, we need to isolate the constant terms on the right side. This involves moving all constant terms from the left side to the right side. In our inequality, 30x + 2 < 8, we can subtract 2 from both sides to move the constant term from the left side to the right side. This gives us 30x + 2 - 2 < 8 - 2, which simplifies to 30x < 6.
Isolating the constant terms is the counterpart to isolating the variable terms. Together, these steps bring us closer to solving for the variable. By keeping the variable terms on one side and the constant terms on the other, we create a clear path towards finding the solution.
Step 5: Solve for the Variable
The final step in solving the inequality is to isolate the variable by dividing both sides by the coefficient of the variable. In our inequality, 30x < 6, the coefficient of x is 30. So, we divide both sides by 30 to isolate x. This gives us 30x / 30 < 6 / 30, which simplifies to x < 1/5. Thus, the solution to the inequality is x < 1/5.
Solving for the variable provides the final answer to the inequality. It tells us the range of values that satisfy the inequality. In this case, any value of x that is less than 1/5 will make the inequality 7x + 2/6 - x < 5x + 4/3 - 4x true. This solution can be represented on a number line or expressed in interval notation, providing a complete picture of the solution set.
Verifying the Solution
After solving an inequality, it’s always a good practice to verify the solution. This helps ensure that no errors were made during the solving process. To verify the solution, we can choose a value from the solution set and substitute it into the original inequality. If the inequality holds true, then our solution is likely correct. We can also choose a value outside the solution set and substitute it into the original inequality. If the inequality does not hold true, this further confirms our solution.
Choosing a Value from the Solution Set
Our solution is x < 1/5. Let’s choose a value from this set, such as x = 0. Substituting x = 0 into the original inequality, 7x + 2/6 - x < 5x + 4/3 - 4x, we get 7(0) + 2/6 - 0 < 5(0) + 4/3 - 4(0), which simplifies to 2/6 < 4/3. This is true, as 2/6 is equal to 1/3, which is less than 4/3. This supports our solution.
Choosing a Value Outside the Solution Set
Now, let’s choose a value outside the solution set, such as x = 1. Substituting x = 1 into the original inequality, 7x + 2/6 - x < 5x + 4/3 - 4x, we get 7(1) + 2/6 - 1 < 5(1) + 4/3 - 4(1), which simplifies to 6 + 1/3 < 1 + 4/3. This is equivalent to 19/3 < 7/3, which is false. This further confirms that our solution x < 1/5 is correct, as the inequality does not hold true for a value outside the solution set.
Conclusion
Solving linear inequalities is a crucial skill in mathematics, with applications in various fields. This article has provided a detailed, step-by-step guide to solving the linear inequality 7x + 2/6 - x < 5x + 4/3 - 4x. We covered the basic principles of solving inequalities, including simplifying both sides, eliminating fractions, isolating variable terms, isolating constant terms, and solving for the variable. We also emphasized the importance of verifying the solution to ensure accuracy.
By following these steps and understanding the underlying principles, you can confidently solve linear inequalities and apply this knowledge to various problem-solving scenarios. Remember to always double-check your work and verify your solution to ensure accuracy. With practice, solving linear inequalities will become second nature, enhancing your mathematical skills and problem-solving abilities.