Solving (t-2)/4 > 11: A Step-by-Step Guide To Finding The Value Of T

In this comprehensive guide, we will delve into the step-by-step solution of the inequality (t-2)/4 > 11. Our primary goal is to determine the range of values for 't' that satisfy this inequality. This article is structured to provide a clear and concise explanation of each step involved, ensuring that readers can easily follow the logic and arrive at the correct answer. Whether you're a student tackling algebra problems or simply seeking to enhance your mathematical skills, this article will equip you with the knowledge to confidently solve similar inequalities.

Understanding Inequalities

Before we dive into the solution, let's establish a clear understanding of inequalities. Inequalities, in mathematics, are used to compare values that are not necessarily equal. Unlike equations, which use the equals sign (=), inequalities use symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). The inequality we are addressing, (t-2)/4 > 11, employs the 'greater than' symbol, indicating that the expression on the left side of the symbol is larger than the value on the right side.

When solving inequalities, the objective is to isolate the variable (in this case, 't') on one side of the inequality symbol. The process is similar to solving equations, with one crucial difference: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed. This is a fundamental rule to remember to ensure accurate solutions.

Step 1: Isolate the Variable Term

The initial step in solving the inequality (t-2)/4 > 11 is to isolate the term containing the variable 't'. Currently, the term (t-2) is being divided by 4. To undo this division, we need to multiply both sides of the inequality by 4. This operation maintains the balance of the inequality, similar to how we would treat an equation. By multiplying both sides by 4, we effectively eliminate the denominator on the left side, bringing us closer to isolating 't'.

Multiplying both sides of the inequality by 4, we get:

4 * [(t-2)/4] > 11 * 4

This simplifies to:

t - 2 > 44

Now, we have a simpler inequality that is easier to work with. The variable term (t-2) is now free from the denominator, and we can proceed to the next step in isolating 't'.

Step 2: Isolate the Variable

Having simplified the inequality to t - 2 > 44, our next goal is to isolate the variable 't' completely. Currently, 't' has a constant term, -2, associated with it. To isolate 't', we need to eliminate this constant term. We can achieve this by performing the inverse operation, which is adding 2 to both sides of the inequality. Just as in the previous step, performing the same operation on both sides ensures that the inequality remains balanced and the relationship between the two sides is maintained.

Adding 2 to both sides of the inequality, we get:

t - 2 + 2 > 44 + 2

This simplifies to:

t > 46

This is our solution! The inequality t > 46 tells us that any value of 't' greater than 46 will satisfy the original inequality (t-2)/4 > 11. We have successfully isolated the variable 't' and determined the range of values that make the inequality true.

Step 3: Verify the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it. We can do this by selecting a value for 't' that falls within the range we found (t > 46) and substituting it back into the original inequality. If the inequality holds true, then our solution is likely correct. Let's choose a value for 't' that is greater than 46; for instance, let's take t = 50.

Substituting t = 50 into the original inequality (t-2)/4 > 11, we get:

(50 - 2) / 4 > 11

Simplifying the expression:

48 / 4 > 11

12 > 11

As we can see, the inequality 12 > 11 is true. This confirms that our solution, t > 46, is correct. We have successfully verified that a value within our solution range satisfies the original inequality, strengthening our confidence in the accuracy of our answer.

The Answer

Therefore, the value of t in the inequality $\frac{t-2}{4} > 11$ is t > 46. So the answer is B. This means that any value of 't' greater than 46 will make the inequality true. We have arrived at this solution through a systematic process of isolating the variable 't', ensuring that each step maintains the balance of the inequality. By following this methodical approach, we can confidently solve a wide range of inequalities.

Conclusion

In this article, we have successfully navigated the process of solving the inequality $\frac{t-2}{4} > 11$. We began by establishing a foundational understanding of inequalities and their significance in mathematics. We then meticulously worked through the steps required to isolate the variable 't', including multiplying both sides of the inequality by 4 and adding 2 to both sides. Each step was carefully explained to ensure clarity and comprehension.

Our journey culminated in the solution t > 46, which signifies that any value of 't' exceeding 46 will satisfy the original inequality. To reinforce the accuracy of our solution, we performed a verification step, substituting t = 50 back into the original inequality and confirming that the resulting statement was indeed true. This verification process underscores the importance of checking one's work in mathematics to ensure correctness.

By mastering the techniques demonstrated in this article, readers can confidently approach and solve similar inequalities. The ability to manipulate inequalities is a valuable skill in algebra and beyond, with applications in various fields such as physics, engineering, and economics. As you continue your mathematical journey, remember the principles of maintaining balance and performing inverse operations, and you'll be well-equipped to tackle a wide array of mathematical challenges.

This article serves as a testament to the power of step-by-step problem-solving and the importance of verifying solutions. By understanding the underlying concepts and applying them methodically, we can unlock the answers to even the most intricate mathematical problems. Whether you're a student, educator, or simply a math enthusiast, the knowledge gained here will undoubtedly prove beneficial in your future endeavors.