Solving The Absolute Value Inequality 6|x-9| >= 12

ge 12$

At the heart of many mathematical problems lies the concept of inequalities, which express the relative size of two values. When combined with absolute value, inequalities can present an interesting challenge. In this comprehensive guide, we will dissect the inequality $6|x-9| ge 12$, providing a step-by-step solution and exploring the underlying principles. Our goal is to clarify the process, making it accessible to learners of all levels. The original problem is: Which of the following is the solution to $6|x-9| ge 12$? A. $x le 7$ or $x ge 11$ B. $x ge 7$ or $x ge 11$ C. $x ge 11$ D. $x le 7$ and $x ge 11$. To solve this, we must delve into the nature of absolute value and how it interacts with inequalities. Absolute value, denoted by vertical bars $| |$, represents the distance of a number from zero on the number line. This means that $|x|$ is equal to $x$ if $x$ is non-negative, and $-x$ if $x$ is negative. This dual nature of absolute value is crucial when solving inequalities involving it. When we encounter an inequality like $|x| > a$, where $a$ is a positive number, we are essentially saying that the distance of $x$ from zero is greater than $a$. This leads to two separate cases: either $x$ is greater than $a$, or $x$ is less than $-a$. Similarly, for $|x| < a$, the solution lies between $-a$ and $a$. Understanding these fundamental principles is the key to unlocking the solution to our given inequality. In the following sections, we will systematically break down the problem, applying these concepts to arrive at the correct answer. By the end of this guide, you'll not only be able to solve this specific problem but also gain a deeper understanding of how to tackle absolute value inequalities in general. This knowledge will prove invaluable in various mathematical contexts, from algebra to calculus and beyond. So, let's embark on this mathematical journey together and unravel the intricacies of this problem. Remember, the key to mastering mathematics is not just about finding the right answer but also about understanding the underlying principles and reasoning.

The first step in solving the inequality $6|x-9| ge 12$ is to isolate the absolute value expression. Isolating absolute value is a fundamental technique in solving inequalities and equations involving absolute values. This isolation simplifies the problem, allowing us to apply the definition of absolute value effectively. To isolate the absolute value, we need to eliminate any coefficients or constants that are multiplied or added to it. In this case, we have the coefficient 6 multiplying the absolute value expression $|x-9|$. To remove this coefficient, we perform the inverse operation, which is division. We divide both sides of the inequality by 6. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains unchanged. Performing this division, we get: $(6|x-9|)/6 ge 12/6$. This simplifies to $|x-9| ge 2$. Now that we have isolated the absolute value expression, the inequality becomes much clearer. The expression $|x-9|$ represents the distance between $x$ and 9 on the number line. The inequality $|x-9| ge 2$ states that this distance is greater than or equal to 2. This understanding is crucial for the next step, where we will split the inequality into two separate cases based on the definition of absolute value. By isolating the absolute value, we have set the stage for a straightforward application of the properties of absolute value inequalities. This step is not just a mechanical manipulation; it's a crucial transformation that reveals the underlying structure of the problem. In the following steps, we will leverage this isolation to solve the inequality and find the solution set for $x$. Remember, each step in solving a mathematical problem is a building block towards the final answer. Isolating the absolute value is a key building block in solving this particular type of inequality.

The essence of absolute value lies in its dual nature: a number's distance from zero can be in either direction. This is where splitting into two cases becomes crucial. Splitting absolute value inequalities into cases is a direct consequence of the definition of absolute value. The absolute value of a quantity is its distance from zero, which means it can be either the quantity itself (if it's non-negative) or the negation of the quantity (if it's negative). In our inequality, $|x-9| ge 2$, the expression inside the absolute value, $x-9$, can be either greater than or equal to zero, or less than zero. This leads to two distinct cases that we must consider separately. Case 1: $x-9 ge 0$. In this case, the absolute value of $x-9$ is simply $x-9$. Therefore, the inequality becomes $x-9 ge 2$. This case represents the situation where $x$ is far enough to the right of 9 on the number line, such that its distance from 9 is at least 2. Case 2: $x-9 < 0$. In this case, the absolute value of $x-9$ is the negation of $x-9$, which is $-(x-9)$. Therefore, the inequality becomes $-(x-9) ge 2$. This case represents the situation where $x$ is far enough to the left of 9 on the number line, such that its distance from 9 is at least 2. By considering these two cases, we ensure that we capture all possible values of $x$ that satisfy the original inequality. Each case represents a different region on the number line, and the solution to the original inequality will be the union of the solutions to these two cases. Splitting into cases is not just a technical step; it's a way of acknowledging the inherent ambiguity in absolute value expressions. It allows us to transform a single inequality involving absolute value into two simpler inequalities that we can solve using standard algebraic techniques. In the next steps, we will solve each of these cases separately, and then combine the solutions to find the complete solution set for the original inequality. This process highlights the power of case-based reasoning in mathematics, a technique that is applicable in many different contexts.

Let's delve into the first scenario we've established: $x - 9 ge 2$. This case represents one part of the solution set for our original inequality. Solving the first case involves isolating $x$ in the inequality $x-9 ge 2$. This is a straightforward algebraic manipulation, but it's a crucial step in finding the overall solution. To isolate $x$, we need to eliminate the constant term, -9, from the left side of the inequality. We do this by performing the inverse operation, which is addition. We add 9 to both sides of the inequality. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains unchanged. Adding 9 to both sides, we get: $x - 9 + 9 ge 2 + 9$. This simplifies to $x ge 11$. This result tells us that all values of $x$ that are greater than or equal to 11 satisfy the inequality in Case 1. On the number line, this corresponds to the region starting at 11 and extending to positive infinity. This is a significant part of the solution, but it's not the whole picture. We still need to consider the second case, which represents the other possible scenario for the absolute value expression. Solving Case 1 is not just about finding a numerical solution; it's about understanding the geometric interpretation of the inequality. The solution $x ge 11$ represents all points on the number line that are at least 2 units away from 9 in the positive direction. This visualization helps to solidify the concept and makes it easier to understand the overall solution when we combine it with the solution from Case 2. In the next step, we will turn our attention to Case 2, solve the corresponding inequality, and then bring together the results from both cases to obtain the final solution set for the original inequality. Remember, each case contributes a portion of the solution, and it's the combination of these portions that gives us the complete answer.

Now, let's tackle the second scenario: $-(x - 9) ge 2$. This case complements the first, capturing the other part of the solution set. Solving the second case requires us to handle the negative sign in front of the parentheses. The inequality $-(x-9) ge 2$ involves a negation, which can be dealt with in a couple of ways. One approach is to distribute the negative sign, and the other is to multiply both sides of the inequality by -1. We'll opt for the latter, as it directly addresses the negative sign and simplifies the inequality. Multiplying both sides by -1, we need to remember a crucial rule: when multiplying or dividing an inequality by a negative number, we must reverse the direction of the inequality sign. This is because multiplying by a negative number flips the order of the numbers on the number line. So, multiplying both sides of $-(x-9) ge 2$ by -1, we get: $(-1) * -(x-9) le (-1) * 2$. This simplifies to $x - 9 le -2$. Now, we have a similar inequality to the one we solved in Case 1. To isolate $x$, we add 9 to both sides of the inequality: $x - 9 + 9 le -2 + 9$. This simplifies to $x le 7$. This result tells us that all values of $x$ that are less than or equal to 7 satisfy the inequality in Case 2. On the number line, this corresponds to the region starting at 7 and extending to negative infinity. Solving Case 2 highlights the importance of paying attention to the details of algebraic manipulations, especially when dealing with inequalities. The rule about reversing the inequality sign when multiplying or dividing by a negative number is a common source of errors, so it's essential to remember it. In the next step, we will combine the solutions from Case 1 and Case 2 to obtain the complete solution set for the original inequality. This will give us a comprehensive understanding of all the values of $x$ that satisfy the given condition.

Having solved both cases, the final step is to weave the solutions together to form the complete answer. Combining the solutions from the two cases involves understanding the logical relationship between them. In our problem, we split the original inequality into two cases based on the definition of absolute value. The solution to the original inequality is the union of the solutions to these two cases. This is because the absolute value condition is satisfied if either one of the cases is true. From Case 1, we found that $x ge 11$. This means that all values of $x$ greater than or equal to 11 satisfy the original inequality. From Case 2, we found that $x le 7$. This means that all values of $x$ less than or equal to 7 also satisfy the original inequality. To combine these solutions, we use the logical