Solving Inequalities A Step-by-Step Guide To 7x + 2 > 30

Linear inequalities are fundamental concepts in algebra, representing relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. In this article, we will delve into the process of solving the linear inequality 7x + 2 > 30, providing a comprehensive step-by-step guide that will empower you to tackle similar problems with confidence. Understanding how to solve inequalities is crucial, as it forms the basis for more advanced mathematical concepts and real-world applications. From determining the range of possible solutions to optimizing resources, inequalities play a vital role in various fields, including economics, engineering, and computer science. So, let's embark on this mathematical journey together and unlock the secrets of solving 7x + 2 > 30.

Understanding the Basics of Inequalities

Before we dive into solving the inequality, it's important to grasp the fundamental concepts of inequalities. Inequalities use symbols like '>' (greater than), '<' (less than), '≥' (greater than or equal to), and '≤' (less than or equal to) to compare two expressions. Unlike equations, which have a single solution, inequalities often have a range of solutions. This range represents all the values that satisfy the inequality. When we talk about solving an inequality, we're essentially trying to isolate the variable (in this case, 'x') on one side of the inequality symbol to determine the set of values that make the inequality true. The techniques used to solve inequalities are similar to those used to solve equations, with one crucial difference: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This is a critical rule to remember, as it directly impacts the solution set. For instance, if we have -x > 2, multiplying both sides by -1 would give us x < -2. Understanding this rule is essential for accurate problem-solving. Let's proceed to our main task: solving 7x + 2 > 30. We will explore the step-by-step process, ensuring you not only arrive at the correct answer but also understand the underlying logic behind each step.

Step 1: Isolating the Variable Term

To begin solving the inequality 7x + 2 > 30, our primary goal is to isolate the term containing the variable, which in this case is 7x. This means we need to get rid of the '+ 2' on the left side of the inequality. To do this, we apply the concept of inverse operations. Since we have addition, we use subtraction. We subtract 2 from both sides of the inequality to maintain the balance. This is a fundamental principle in solving both equations and inequalities: whatever operation you perform on one side, you must perform on the other side to preserve the relationship. Subtracting 2 from both sides, we get: 7x + 2 - 2 > 30 - 2. Simplifying this gives us 7x > 28. Now, we have successfully isolated the variable term, 7x, on the left side. This is a significant step forward because it brings us closer to isolating the variable 'x' itself. By isolating the variable term, we pave the way for the next step, which involves isolating the variable 'x' completely. Remember, each step we take is aimed at simplifying the inequality and revealing the solution set. So, with 7x > 28, we are now in a position to isolate 'x' and determine the range of values that satisfy the original inequality.

Step 2: Isolating the Variable

Having successfully isolated the variable term in the previous step, we now focus on isolating the variable 'x' completely. Our inequality currently reads 7x > 28. This means 7 multiplied by 'x' is greater than 28. To isolate 'x', we need to undo the multiplication. The inverse operation of multiplication is division, so we will divide both sides of the inequality by 7. It's crucial to remember the rule about inequalities: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. However, in this case, we are dividing by a positive number (7), so we do not need to reverse the inequality sign. Dividing both sides by 7, we get: (7x) / 7 > 28 / 7. Simplifying this, we have x > 4. This is our solution! It tells us that any value of 'x' greater than 4 will satisfy the original inequality 7x + 2 > 30. We have now successfully isolated 'x' and determined the solution set. Understanding this step is vital, as it's a common technique used in solving various types of inequalities. The key is to identify the operation being performed on the variable and then apply the inverse operation to both sides of the inequality.

Step 3: Interpreting the Solution

After performing the algebraic steps, we've arrived at the solution x > 4. But what does this actually mean? Interpreting the solution correctly is as important as solving the inequality itself. The solution x > 4 tells us that 'x' can be any number that is strictly greater than 4. It does not include 4 itself. This is a crucial distinction because if the solution were x ≥ 4, it would mean 'x' can be 4 or any number greater than 4. To visualize this, we can think of a number line. Imagine a number line stretching from negative infinity to positive infinity. The solution x > 4 would be represented by an open circle at 4 (indicating that 4 is not included) and an arrow extending to the right, indicating all numbers greater than 4. Examples of numbers that satisfy the inequality are 4.0001, 5, 10, 100, or any other number you can think of that is larger than 4. Numbers like 3, 4, and -1 do not satisfy the inequality because they are not greater than 4. Understanding how to interpret the solution is vital for applying this knowledge to real-world scenarios. For example, if this inequality represented a constraint in a problem, we would know that the variable 'x' must be a value greater than 4 to meet that constraint. So, the solution x > 4 is not just a mathematical result; it's a piece of information that can be used to make decisions and solve problems in various contexts.

Step 4: Verifying the Solution

Once we've found a solution to an inequality, it's always a good practice to verify our answer. This helps ensure that we haven't made any errors in our calculations and that our solution is indeed correct. To verify the solution x > 4 for the inequality 7x + 2 > 30, we can choose a value of 'x' that is greater than 4 and substitute it back into the original inequality. If the inequality holds true, then our solution is likely correct. Let's choose x = 5 as our test value, since 5 is clearly greater than 4. Substituting x = 5 into the original inequality, we get: 7(5) + 2 > 30. Simplifying, we have 35 + 2 > 30, which further simplifies to 37 > 30. This statement is true! Since 37 is indeed greater than 30, our chosen value of x = 5 satisfies the original inequality. This gives us confidence that our solution x > 4 is correct. However, one test value is not definitive proof. To be absolutely sure, we could also test a value that is not in the solution set (e.g., x = 4) and confirm that it does not satisfy the inequality. If we substitute x = 4, we get 7(4) + 2 > 30, which simplifies to 30 > 30. This statement is false, confirming that 4 is not part of the solution set. By verifying our solution, we strengthen our confidence in the accuracy of our work and ensure that we have a solid understanding of the problem.

Conclusion: Mastering Linear Inequalities

In conclusion, solving the linear inequality 7x + 2 > 30 involves a systematic approach of isolating the variable and interpreting the solution. We began by understanding the basic principles of inequalities, recognizing the symbols and their meanings. We then moved on to the step-by-step process of solving the inequality: first, isolating the variable term by subtracting 2 from both sides, and then isolating the variable 'x' by dividing both sides by 7. This led us to the solution x > 4, which means any value of 'x' greater than 4 will satisfy the original inequality. We emphasized the importance of interpreting this solution, visualizing it on a number line, and understanding what it means in the context of a problem. Finally, we highlighted the crucial step of verifying the solution by substituting a value from our solution set back into the original inequality to ensure its accuracy. By mastering these steps, you can confidently tackle a wide range of linear inequalities. The skills you've gained here are not just applicable to math problems; they extend to various real-world scenarios where understanding ranges and constraints is essential. So, continue practicing and building your understanding of inequalities, and you'll find yourself well-equipped to solve more complex mathematical challenges.

Therefore, the correct answer is C. x > 4