Solving Inequalities A Step-by-Step Guide To Kendall's Equation

Inequalities can seem like complex puzzles, but with a systematic approach, they become quite manageable. In mathematics, inequalities are statements that compare two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving inequalities involves isolating the variable on one side, much like solving equations, but with an added consideration for how certain operations affect the inequality sign. When dealing with inequalities, the primary goal is to isolate the variable to determine the range of values that satisfy the given condition. This process typically involves performing inverse operations to undo the operations applied to the variable, all while maintaining the balance of the inequality. Think of an inequality as a balance scale, where the expressions on each side must remain balanced. Any operation performed on one side must also be performed on the other to maintain this balance. The first step in solving an inequality is often the most crucial, as it sets the stage for the subsequent steps. The order in which operations are performed can significantly impact the ease and efficiency of the solution process. Understanding the structure of the inequality and identifying the operations that need to be undone are key skills in mastering inequality solving. In Kendall’s case, the inequality presents a specific order of operations that must be addressed to isolate the variable 'x'. By carefully analyzing the inequality, we can determine the most strategic first step to simplify the problem and move closer to the solution. Let's dive into the specific inequality Kendall is tackling and explore the logical steps to solve it efficiently. Let's consider Kendall's challenge.

Understanding the Inequality: x / 2 + 6 > 42

Kendall is faced with the inequality x / 2 + 6 > 42. To effectively solve this, we need to understand the structure of the inequality and the operations involved. The inequality states that the expression 'x / 2 + 6' is greater than 42. This means that any value of 'x' that satisfies this condition will be a solution to the inequality. To find these values, we need to isolate 'x' on one side of the inequality. Isolating 'x' involves undoing the operations that are applied to it. In this case, 'x' is first divided by 2, and then 6 is added to the result. To isolate 'x', we need to reverse these operations in the opposite order. This is a fundamental principle in solving both equations and inequalities. By understanding the order of operations and applying inverse operations, we can systematically simplify the inequality. The order of operations dictates that multiplication and division are performed before addition and subtraction. However, when solving for 'x', we must reverse this order. This means we should address the addition of 6 before dealing with the division by 2. This approach will help us efficiently isolate 'x' and determine the solution set for the inequality. Recognizing the structure of the inequality is the first step towards solving it. Understanding the operations applied to the variable and the order in which they were applied allows us to strategically plan the steps needed to isolate the variable. In the next section, we will discuss the possible first steps Kendall can take and evaluate their effectiveness in solving the inequality.

Identifying the Correct First Step

When solving the inequality x / 2 + 6 > 42, Kendall has several options for the first step, but only one is the most efficient. Let's evaluate the given options:

• A. Subtract 6 from both sides of the inequality: This is a valid and often the most strategic first step in this type of inequality. Subtracting 6 from both sides begins the process of isolating 'x' by undoing the addition operation. This maintains the balance of the inequality while simplifying the expression. This approach aligns with the principle of reversing the order of operations, making it a logical and effective starting point.
• B. Divide both sides of the inequality by 2: Dividing by 2 at this stage would complicate the equation. It would require distributing the division across both terms on the left side, leading to fractions and potentially increasing the difficulty of the problem. While mathematically valid, this step is less efficient and can make the solving process more prone to errors. The goal is to simplify the inequality, and this step would move in the opposite direction.
• C. Add 6 to both sides of the inequality: Adding 6 to both sides would undo a subtraction operation, but since the operation in the original inequality is addition, this step would not help in isolating 'x'. It would further complicate the inequality by increasing the constant term on the right side. This option is counterproductive and does not align with the strategy of isolating the variable.

By carefully considering each option, it becomes clear that subtracting 6 from both sides is the most logical first step. This approach directly addresses the addition of 6 on the left side, moving us closer to isolating 'x'. It also keeps the numbers manageable and avoids unnecessary complications. The choice of the first step can significantly impact the ease and efficiency of solving an inequality. A strategic first step sets the stage for a smoother solution process, while a less effective choice can lead to unnecessary complexity.

The Strategic Advantage of Subtracting 6

Subtracting 6 from both sides of the inequality x / 2 + 6 > 42 is the most strategic first step because it directly addresses the addition operation, moving us closer to isolating the variable 'x'. When we subtract 6 from both sides, we are essentially undoing the addition of 6 on the left side of the inequality. This simplifies the inequality and makes it easier to work with. The resulting inequality is x / 2 > 36, which is a much simpler form than the original. This strategic move reduces the complexity of the problem and sets the stage for the next step in solving for 'x'. The advantage of this approach lies in its directness and efficiency. By focusing on undoing the addition, we avoid introducing unnecessary operations or complications. This aligns with the principle of simplifying expressions before performing more complex operations. Keeping the numbers manageable is another benefit of subtracting 6 first. This approach avoids dealing with fractions or larger numbers prematurely, which can reduce the likelihood of errors. It also allows us to maintain a clear and organized solution process. Furthermore, subtracting 6 first follows the reverse order of operations. In the original inequality, 'x' is first divided by 2, and then 6 is added. To isolate 'x', we must reverse this order, meaning we address the addition before the division. This is a fundamental concept in solving equations and inequalities. By choosing this strategic first step, Kendall sets herself up for a more straightforward and accurate solution to the inequality.

Step-by-Step Solution After the First Move

After Kendall subtracts 6 from both sides of the inequality x / 2 + 6 > 42, the inequality becomes x / 2 > 36. This simplified form makes it much easier to isolate 'x'. The next step is to undo the division by 2. To do this, we multiply both sides of the inequality by 2. This is the inverse operation of division and will effectively isolate 'x' on the left side. Multiplying both sides by 2 gives us: (x / 2) * 2 > 36 * 2. This simplifies to x > 72. This inequality tells us that 'x' must be greater than 72 to satisfy the original condition. Any value of 'x' greater than 72 will make the expression 'x / 2 + 6' greater than 42. We can check this solution by substituting a value greater than 72 into the original inequality. For example, if we let x = 74, we have: 74 / 2 + 6 > 42. This simplifies to 37 + 6 > 42, which is 43 > 42. This is a true statement, confirming that our solution is correct. The step-by-step process of solving this inequality demonstrates the importance of strategic thinking and applying inverse operations in the correct order. By subtracting 6 first, we simplified the inequality and made it easier to isolate 'x'. Then, by multiplying by 2, we completed the process and found the solution. This systematic approach is key to solving inequalities effectively and accurately. Understanding each step and its purpose is crucial for mastering inequality solving.

Conclusion: Mastering Inequality Solutions

In conclusion, when solving the inequality x / 2 + 6 > 42, the most effective first step for Kendall is to subtract 6 from both sides of the inequality (Option A). This strategic move simplifies the inequality and sets the stage for a straightforward solution. By subtracting 6, Kendall undoes the addition operation, bringing her closer to isolating the variable 'x'. The subsequent step involves multiplying both sides by 2 to completely isolate 'x', revealing the solution x > 72. This process highlights the importance of understanding the order of operations and applying inverse operations in the correct sequence. Mastering inequality solutions requires a systematic approach, strategic thinking, and a clear understanding of the underlying principles. By carefully analyzing the inequality and choosing the most effective first step, we can simplify the problem and arrive at the correct solution efficiently. Inequalities are a fundamental concept in mathematics, and the ability to solve them is essential for various applications in algebra and beyond. By practicing and applying these strategies, students can develop confidence and proficiency in solving inequalities of all types. The key takeaway is that the first step can significantly impact the ease and accuracy of the solution process. Choosing the right first step is a critical skill in mastering inequality solutions.