Solving Inequalities A Step-by-Step Guide To 38 < 4x + 3 + 7 - 3x

Solving inequalities is a fundamental concept in mathematics, extending the principles of solving equations to situations where values are not necessarily equal but rather fall within a certain range. Unlike equations, which have specific solutions, inequalities define a range of values that satisfy a given condition. This article delves into the process of solving the inequality $38 < 4x + 3 + 7 - 3x$, providing a detailed, step-by-step explanation to enhance understanding and problem-solving skills in this area of mathematics. Inequalities, at their core, involve comparing mathematical expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols create a relationship between two expressions, indicating that one is either smaller, larger, or within a certain bound compared to the other. Understanding these symbols is crucial for interpreting and solving inequalities accurately. When we talk about solving inequalities, we're essentially looking for all the values of the variable that make the inequality statement true. This is similar to solving equations, but with a key difference: the solution to an inequality is often a range of values, rather than a single value. This range is defined by the inequality symbol and the mathematical operations performed on the inequality. For example, if we have the inequality $x > 5$, the solution is any value of x that is greater than 5. This could be 5.1, 6, 10, or any other number that fits this condition. The process of solving inequalities involves manipulating the inequality using algebraic operations, just like solving equations. However, there's a crucial rule to keep in mind: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the values, and thus the relationship between the two sides of the inequality. In the context of real-world problems, inequalities are incredibly useful. They allow us to model situations where there are constraints or ranges of possibilities. For example, we might use an inequality to represent the maximum number of items we can purchase within a budget, or the minimum score we need to achieve on a test to pass. Understanding how to solve and interpret inequalities is therefore a valuable skill in many areas of life. As we delve into the specific inequality $38 < 4x + 3 + 7 - 3x$, we'll see how these principles are applied in practice. By breaking down the problem step by step, we can gain a clearer understanding of the process and the logic behind it. This will not only help in solving this particular inequality but also in tackling other similar problems in the future. The ability to confidently solve inequalities is a cornerstone of mathematical literacy and is essential for success in more advanced mathematical topics. So, let's embark on this journey to unlock the secrets of inequalities and master the art of solving them.

Step 1: Simplify the Inequality

The first key step in solving the inequality $38 < 4x + 3 + 7 - 3x$ is to simplify both sides of the inequality. This involves combining like terms to make the expression more manageable and easier to work with. Simplifying an inequality is similar to simplifying an equation; the goal is to reduce the number of terms and make the inequality clearer. In this case, we focus on combining the constant terms on the right side of the inequality and then combining the terms involving the variable $x$. The right side of the inequality is $4x + 3 + 7 - 3x$. We can see that there are two constant terms, 3 and 7, and two terms involving $x$, $4x$ and $-3x$. The first step in simplifying is to combine the constant terms. Adding 3 and 7 gives us 10. So, the expression becomes $4x + 10 - 3x$. Next, we combine the terms involving $x$. We have $4x$ and $-3x$. When we add these together, we get $4x - 3x$, which simplifies to $1x$ or simply $x$. Therefore, the right side of the inequality, $4x + 3 + 7 - 3x$, simplifies to $x + 10$. Now, we rewrite the entire inequality with the simplified right side. The original inequality was $38 < 4x + 3 + 7 - 3x$. After simplifying the right side, the inequality becomes $38 < x + 10$. This simplified form is much easier to work with. It allows us to see more clearly what needs to be done to isolate the variable $x$ and solve the inequality. Simplifying the inequality is a crucial step because it reduces the complexity of the problem. By combining like terms, we make the inequality more concise and easier to understand. This, in turn, makes the subsequent steps of solving the inequality less prone to errors. A simplified inequality allows us to focus on the essential part of the problem: isolating the variable. Without simplification, the inequality might look intimidating and difficult to solve. But by breaking it down into smaller, more manageable parts, we can approach the problem with confidence. This step of simplifying the inequality is not just a mathematical trick; it's a way of organizing our thoughts and making the problem more accessible. It's a fundamental skill in algebra and is used extensively in solving various types of equations and inequalities. So, by mastering this step, we're building a strong foundation for more advanced mathematical concepts. In summary, the first step in solving the inequality $38 < 4x + 3 + 7 - 3x$ is to simplify it by combining like terms. This results in the simplified inequality $38 < x + 10$, which is a crucial step towards isolating the variable $x$ and finding the solution to the inequality.

Step 2: Isolate the Variable

After simplifying the inequality to $38 < x + 10$, the next crucial step in solving for x is to isolate the variable $x$ on one side of the inequality. Isolating the variable means getting $x$ by itself on either the left or the right side of the inequality. This is achieved by performing operations that undo any addition or subtraction affecting the variable. In this case, we have $x + 10$ on the right side of the inequality. To isolate $x$, we need to eliminate the $+10$. The way to do this is to subtract 10 from both sides of the inequality. This is a fundamental principle in solving inequalities: whatever operation you perform on one side, you must also perform on the other side to maintain the balance of the inequality. Subtracting 10 from both sides of $38 < x + 10$ gives us: $38 - 10 < x + 10 - 10$. Now, we simplify both sides. On the left side, $38 - 10$ equals 28. On the right side, $10 - 10$ cancels out, leaving us with just $x$. So, the inequality becomes $28 < x$. This inequality tells us that 28 is less than $x$, which is the same as saying that $x$ is greater than 28. We have now successfully isolated the variable $x$. The process of isolating the variable is a cornerstone of solving inequalities and equations. It's a method that allows us to peel away the layers of operations that are affecting the variable, one by one, until we are left with the variable alone on one side. This gives us a clear understanding of the values that the variable can take. It's important to note that when we add or subtract the same number from both sides of an inequality, the direction of the inequality sign remains the same. This is because adding or subtracting a number shifts both sides of the inequality by the same amount, without changing their relative positions. However, as mentioned earlier, multiplying or dividing by a negative number requires us to flip the inequality sign, which is not the case in this particular problem. Isolating the variable is not just a mechanical process; it's a way of revealing the underlying relationship between the variable and the constants in the inequality. It helps us to understand the range of values that satisfy the inequality and to express the solution in a clear and concise way. In summary, to isolate the variable $x$ in the inequality $38 < x + 10$, we subtract 10 from both sides, resulting in the inequality $28 < x$. This means that any value of $x$ that is greater than 28 will satisfy the original inequality. This step is crucial in determining the solution set for the inequality and understanding the range of possible values for $x$.

Step 3: Interpret the Solution

After isolating the variable in the inequality $38 < 4x + 3 + 7 - 3x$, we arrived at the solution $28 < x$. This is a crucial step, but equally important is the interpretation of this solution. Understanding what $28 < x$ means is essential for applying the solution correctly and in the right context. The inequality $28 < x$ states that 28 is less than $x$. In simpler terms, this means that $x$ is greater than 28. The '<' symbol here is the 'less than' symbol, and it indicates that the value on the left (28) is smaller than the value on the right ($x$). So, any number that is larger than 28 will satisfy the original inequality. To fully grasp this, it's helpful to think of a number line. If you were to plot this inequality on a number line, you would find 28 marked as a boundary point. Since $x$ is greater than 28, you would shade the portion of the number line to the right of 28. This shaded area represents all the possible values of $x$ that make the inequality true. It's important to note that 28 itself is not included in the solution set. This is because the inequality is strictly 'less than' ($<$), not 'less than or equal to' ($\\,leq$). If the inequality were $28 \leq x$, then 28 would be included in the solution set. The solution $28 < x$ represents an infinite number of possible values for $x$. For example, $x$ could be 28.1, 29, 30, 100, or even 1000. As long as the value is greater than 28, it satisfies the inequality. This is a key difference between solutions to inequalities and solutions to equations. Equations typically have one or a few specific solutions, while inequalities often have a range of solutions. Interpreting the solution also involves understanding the context of the problem. In a real-world scenario, the solution might have practical implications. For example, if $x$ represents the number of items you need to sell to make a profit, then the solution $28 < x$ would mean you need to sell more than 28 items. It's not enough to just find the solution; you need to understand what it means in the context of the situation. Understanding the solution set is also crucial for checking the answer. You can pick a value from the solution set and plug it back into the original inequality to see if it holds true. For example, if we choose $x = 30$, which is greater than 28, and substitute it into the original inequality $38 < 4x + 3 + 7 - 3x$, we can verify that it indeed satisfies the inequality. In summary, interpreting the solution $28 < x$ involves understanding that $x$ is any number greater than 28. This means that the solution set includes all values to the right of 28 on a number line, excluding 28 itself. This interpretation is essential for applying the solution in practical situations and for verifying the correctness of the solution. The ability to accurately interpret solutions to inequalities is a crucial skill in mathematics and in real-world problem-solving.

Step 4: Select the Correct Answer

After solving the inequality $38 < 4x + 3 + 7 - 3x$ and interpreting the solution as $x > 28$, the final step is to identify the correct answer from the given options. This step requires a careful comparison of the derived solution with the provided choices to ensure that the correct option is selected. The given options are:

A. $x > 28$ B. $x > 4$ C. $x < 28$ D. $x < 4$

By comparing our solution, $x > 28$, with the given options, it becomes clear that option A, $x > 28$, is the correct answer. This option perfectly matches the solution we derived through the step-by-step process of simplifying, isolating the variable, and interpreting the result. It is important to note why the other options are incorrect. Option B, $x > 4$, suggests that $x$ is greater than 4. While this is a range of values, it does not accurately represent the solution to the given inequality, as it includes values less than or equal to 28, which do not satisfy the original inequality. Option C, $x < 28$, indicates that $x$ is less than 28. This is the opposite of the correct solution. This option includes values that are smaller than 28, which would not make the original inequality true. Option D, $x < 4$, suggests that $x$ is less than 4. This is also incorrect because the solution to the inequality requires $x$ to be greater than 28, a completely different range of values. Selecting the correct answer is not just about finding a match; it's about confirming that the chosen option accurately reflects the entire solution process. Each step, from simplifying to isolating the variable, leads to the final solution, and the selected answer should be a direct representation of that solution. This final step is a crucial checkpoint in the problem-solving process. It ensures that all the previous steps have been performed correctly and that the solution is accurately represented. By carefully comparing the derived solution with the given options, we minimize the risk of errors and ensure that the correct answer is chosen. In the context of assessments or exams, this step is particularly important. It is the culmination of the problem-solving effort and determines whether the question is answered correctly. Therefore, taking the time to carefully compare the solution with the options is a worthwhile investment. In summary, after solving the inequality $38 < 4x + 3 + 7 - 3x$ and arriving at the solution $x > 28$, the correct answer from the given options is A. $x > 28$. This step involves a careful comparison of the derived solution with the provided choices, ensuring that the selected answer accurately represents the solution obtained through the problem-solving process.

In conclusion, solving the inequality $38 < 4x + 3 + 7 - 3x$ involves a series of logical steps, each building upon the previous one to arrive at the correct solution. This process not only provides the answer but also enhances understanding of inequality principles and problem-solving techniques. The initial step was to simplify the inequality by combining like terms. This made the inequality more manageable and set the stage for isolating the variable. Simplifying is a fundamental skill in algebra, allowing us to reduce complex expressions into more understandable forms. Next, we isolated the variable $x$ by performing the necessary operations to get $x$ alone on one side of the inequality. This involved subtracting 10 from both sides, maintaining the balance of the inequality. The ability to isolate variables is a cornerstone of algebraic manipulation and is crucial for solving a wide range of mathematical problems. After isolating the variable, we arrived at the solution $x > 28$. This solution tells us that any value of $x$ greater than 28 will satisfy the original inequality. However, finding the solution is only part of the process; interpreting the solution is equally important. Understanding what $x > 28$ means in the context of the problem is essential for applying the solution correctly. Finally, we compared the derived solution with the given options and identified the correct answer as A. $x > 28$. This step is a critical checkpoint, ensuring that the chosen answer accurately reflects the solution obtained through the entire problem-solving process. The process of solving this inequality highlights several key principles in mathematics. It demonstrates the importance of simplifying expressions, the rules for manipulating inequalities, and the significance of interpreting solutions. These principles are not only applicable to this specific problem but are also foundational for more advanced mathematical concepts. Furthermore, this exercise reinforces the value of a systematic approach to problem-solving. By breaking down the problem into smaller, manageable steps, we can tackle complex problems with confidence and accuracy. Each step, from simplifying to selecting the correct answer, contributes to the overall solution and enhances our understanding of the underlying mathematical concepts. In addition to the mathematical aspects, solving inequalities also develops critical thinking skills. It requires us to analyze the problem, identify the relevant information, and apply the appropriate techniques to find the solution. These skills are transferable to various areas of life, making the study of mathematics a valuable endeavor. In summary, solving the inequality $38 < 4x + 3 + 7 - 3x$ is more than just finding the correct answer. It's a journey through the principles of algebra, a demonstration of problem-solving techniques, and an exercise in critical thinking. By mastering these skills, we not only become proficient in mathematics but also develop a valuable set of tools for tackling challenges in any field.

Therefore, the correct answer is A. $x > 28$