Solve The Inequality $12p + 7 > 139$. The Options Are A. $p > 12$, B. $p > 18$, C. $p > 11$, D. $p < 11$.

Solving inequalities is a fundamental concept in algebra, and it's crucial to grasp the underlying principles to tackle more complex mathematical problems. An inequality, unlike an equation, doesn't state that two expressions are equal; instead, it indicates a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. This distinction leads to a slightly different approach when solving inequalities compared to equations, particularly when dealing with multiplication or division by negative numbers. Before diving into the specific problem, let's solidify our understanding of the basics. Inequalities are represented using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). When we solve an inequality, our goal is to isolate the variable on one side, just as we do with equations. However, a key difference arises when we multiply or divide both sides of an inequality by a negative number. In such cases, we must flip the direction of the inequality sign to maintain the truth of the statement. This is because multiplying or dividing by a negative number reverses the order of numbers on the number line. For example, if we have -2 < 4, multiplying both sides by -1 gives us 2 > -4, demonstrating the necessity of flipping the sign. With these foundational concepts in mind, we're well-equipped to tackle the given inequality and find the solution set. Remember, the solution to an inequality is often a range of values, rather than a single value, which adds another layer of understanding to the process.

Our primary focus here is deconstructing the inequality $12p + 7 > 139$ to isolate the variable p. This process involves applying algebraic operations to both sides of the inequality while maintaining its balance. The ultimate goal is to arrive at a statement of the form p > some value, which will represent the solution set. The given inequality, $12p + 7 > 139$, presents a linear relationship between p and a constant. To solve for p, we need to undo the operations that are being applied to it. The first step in this process is to isolate the term containing p, which in this case is $12p$. We can achieve this by subtracting 7 from both sides of the inequality. This operation maintains the balance of the inequality and moves us closer to isolating p. Subtracting 7 from both sides, we get: $12p + 7 - 7 > 139 - 7$, which simplifies to $12p > 132$. Now, we have a simpler inequality where $12p$ is greater than 132. The next step is to isolate p completely by undoing the multiplication by 12. This is accomplished by dividing both sides of the inequality by 12. Since 12 is a positive number, we don't need to worry about flipping the inequality sign. Dividing both sides by 12, we have: $(12p) / 12 > 132 / 12$, which simplifies to $p > 11$. This final inequality represents the solution set for the original inequality. It tells us that any value of p greater than 11 will satisfy the condition $12p + 7 > 139$. We have successfully deconstructed the inequality and found the range of values for p that make the statement true. This step-by-step approach ensures accuracy and clarity in the solution process.

To effectively isolate the variable p in the inequality $12p + 7 > 139$, we follow a systematic, step-by-step approach. This method not only provides the correct solution but also enhances understanding of the underlying algebraic principles. The initial inequality, $12p + 7 > 139$, has two operations acting on p: multiplication by 12 and addition of 7. To isolate p, we must reverse these operations in the correct order. According to the order of operations, we typically address addition and subtraction before multiplication and division. Therefore, our first step is to eliminate the addition of 7. We achieve this by subtracting 7 from both sides of the inequality. This ensures that we maintain the balance of the inequality, as whatever operation we perform on one side, we must also perform on the other side. Subtracting 7 from both sides, we get: $12p + 7 - 7 > 139 - 7$, which simplifies to $12p > 132$. Now, we have successfully isolated the term containing p on the left side. The next step is to undo the multiplication by 12. To do this, we divide both sides of the inequality by 12. It's crucial to remember that when we multiply or divide an inequality by a negative number, we must flip the inequality sign. However, in this case, we are dividing by a positive number (12), so the inequality sign remains unchanged. Dividing both sides by 12, we have: $(12p) / 12 > 132 / 12$, which simplifies to $p > 11$. This final inequality, $p > 11$, represents the solution to the original inequality. It tells us that any value of p greater than 11 will satisfy the given condition. By following this step-by-step process, we have successfully isolated the variable and found the solution set.

After systematically solving the inequality $12p + 7 > 139$, the crucial step is identifying the correct answer from the given options. This involves carefully comparing our solution with the provided choices and selecting the one that accurately represents the range of values for p. Our step-by-step solution led us to the inequality $p > 11$. This inequality states that p can be any value greater than 11, but it cannot be equal to 11. Now, let's examine the provided options:

A. $p > 12$ B. $p > 18$ C. $p > 11$ D. $p < 11$

Comparing our solution, $p > 11$, with the options, we can see that option C, $p > 11$, perfectly matches our result. This means that any value of p greater than 11 will satisfy the original inequality. Options A and B, $p > 12$ and $p > 18$, respectively, represent subsets of the solution set, but they are not the complete solution. For example, a value like 11.5 would satisfy $p > 11$ but not $p > 12$ or $p > 18$. Option D, $p < 11$, represents the opposite of the solution set, indicating values of p that are less than 11, which would not satisfy the original inequality. Therefore, the correct answer is unequivocally option C, $p > 11$. This careful comparison and selection process ensures that we not only solve the inequality correctly but also accurately identify the corresponding answer from the given choices.

In conclusion, mastering the solution of inequalities is a vital skill in algebra and mathematics in general. The process involves a clear understanding of the principles of inequalities, careful application of algebraic operations, and precise identification of the solution set. By systematically isolating the variable, we can determine the range of values that satisfy the given inequality. In the specific example of $12p + 7 > 139$, we demonstrated a step-by-step approach: first, subtracting 7 from both sides to isolate the term with p, and then dividing both sides by 12 to isolate p itself. This led us to the solution $p > 11$, which means any value of p greater than 11 will make the inequality true. It's crucial to remember the key difference between solving equations and inequalities, particularly when multiplying or dividing by negative numbers, which requires flipping the inequality sign. This understanding is fundamental to avoiding errors and arriving at the correct solution. Furthermore, the ability to compare the solution with given options and accurately identify the correct answer is an essential skill in problem-solving. Inequalities appear in various mathematical contexts and real-world applications, from optimization problems to modeling constraints. Therefore, a solid grasp of inequality solutions is not only beneficial for academic success but also for practical problem-solving in various fields. By practicing and applying these principles, one can confidently tackle a wide range of inequality problems.