Solving Absolute Value Inequalities: |x-6| ≥ 6

Hey guys! Today, we're diving into the fascinating world of absolute value inequalities. Specifically, we're going to tackle the inequality |x-6| ≥ 6 and solve it algebraically. This type of problem might seem intimidating at first, but don't worry! We'll break it down step-by-step so it becomes crystal clear. Understanding how to solve absolute value inequalities is a crucial skill in algebra and can pop up in various real-world applications, from engineering to economics. So, grab your pencils, and let's get started!

Understanding Absolute Value

Before we jump into solving the inequality, let's quickly recap what absolute value actually means. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. For example, the absolute value of 5, written as |5|, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, written as |-5|, is also 5 because -5 is also 5 units away from zero. Think of it like this: absolute value strips away the sign, leaving you with just the magnitude.

Now, why is this important for inequalities? Well, when we have an absolute value inequality like |x-6| ≥ 6, it means that the expression x-6 is either 6 units away from zero in the positive direction or 6 units away from zero in the negative direction (or even further away!). This leads to two separate cases that we need to consider to find all possible values of x that satisfy the inequality. Remember, the absolute value ensures we are always dealing with a positive distance, but the expression inside the absolute value can indeed be negative.

Understanding this fundamental concept is key to correctly setting up and solving the two cases that arise from the absolute value inequality. Without a solid grasp of what absolute value represents, it's easy to get confused about why we need to split the problem into two separate scenarios. So, keep this definition in mind as we move forward – it will be your guiding star!

Breaking Down the Inequality

The absolute value inequality |x-6| ≥ 6 tells us that the distance between x and 6 is greater than or equal to 6. This gives us two possible scenarios:

1. Scenario 1: The expression inside the absolute value is positive or zero. In this case, x-6 is already greater than or equal to 0, so we can simply remove the absolute value signs and solve the inequality x-6 ≥ 6. This scenario represents the situation where x is to the right of 6 on the number line.
2. Scenario 2: The expression inside the absolute value is negative. In this case, x-6 is less than 0, so we need to take the opposite of the expression inside the absolute value to make it positive. This means we solve the inequality -(x-6) ≥ 6. This scenario represents the situation where x is to the left of 6 on the number line.

By considering both scenarios, we cover all possible values of x that satisfy the original absolute value inequality. It's like saying, "Hey, x could be far enough to the right of 6, or it could be far enough to the left of 6!" We need to explore both possibilities to get the complete picture. This splitting into cases is the hallmark of solving absolute value inequalities, and mastering this technique is crucial for tackling more complex problems later on. Don't be tempted to only consider one case – always remember to account for both the positive and negative possibilities that the absolute value introduces!

Solving Scenario 1: x - 6 ≥ 6

Let's solve the first scenario: x - 6 ≥ 6. This is a straightforward linear inequality. To isolate x, we need to get rid of the -6 on the left side. We can do this by adding 6 to both sides of the inequality. Remember, whatever you do to one side of an inequality, you must do to the other side to maintain the balance. So, we have:

x - 6 + 6 ≥ 6 + 6

This simplifies to:

x ≥ 12

So, in the first scenario, x must be greater than or equal to 12. This means any value of x that is 12 or larger will satisfy the original inequality. For example, if x = 15, then |15 - 6| = |9| = 9, which is indeed greater than or equal to 6. This solution represents all the numbers on the number line that are to the right of 12, including 12 itself. We can visualize this as a closed interval on the number line, starting at 12 and extending to positive infinity. Keep this result in mind as we move on to solving the second scenario – we'll combine the solutions from both scenarios at the end to get the complete solution set for the original inequality. It's like putting together the pieces of a puzzle!

Solving Scenario 2: -(x - 6) ≥ 6

Now, let's tackle the second scenario: -(x - 6) ≥ 6. This inequality looks a little different because of the negative sign in front of the parentheses. Our first step is to distribute the negative sign to both terms inside the parentheses:

-x + 6 ≥ 6

Now, we want to isolate x. Let's subtract 6 from both sides of the inequality:

-x + 6 - 6 ≥ 6 - 6

This simplifies to:

-x ≥ 0

But we're not quite done yet! We want to solve for x, not -x. To get rid of the negative sign on x, we can multiply both sides of the inequality by -1. However, there's a crucial rule to remember: when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. So, when we multiply both sides by -1, the ≥ sign becomes a ≤ sign:

(-1)(-x) ≤ (-1)(0)

This simplifies to:

x ≤ 0

So, in the second scenario, x must be less than or equal to 0. This means any value of x that is 0 or smaller will satisfy the original inequality. For example, if x = -3, then |-3 - 6| = |-9| = 9, which is indeed greater than or equal to 6. This solution represents all the numbers on the number line that are to the left of 0, including 0 itself. We can visualize this as a closed interval on the number line, starting at negative infinity and extending to 0. Now that we've solved both scenarios, it's time to combine the results and get the complete solution!

Combining the Solutions

We've found that x ≥ 12 in the first scenario and x ≤ 0 in the second scenario. To get the complete solution to the original inequality |x-6| ≥ 6, we need to combine these two solutions. This means that x can either be greater than or equal to 12, or x can be less than or equal to 0. There's no overlap between these two intervals, so we simply list them separately.

We can express the solution in interval notation as:

(-∞, 0] ∪ [12, ∞)

This notation means that the solution includes all numbers from negative infinity up to and including 0, as well as all numbers from 12 up to positive infinity. The ∪ symbol represents the union of the two intervals, meaning we're combining them to form the complete solution set. In plain English, this means that any number that is either less than or equal to 0 or greater than or equal to 12 will satisfy the original inequality. Isn't that neat?

Visualizing the Solution on a Number Line

To really solidify our understanding, let's visualize the solution on a number line. Draw a number line and mark the points 0 and 12. Since our solution includes values less than or equal to 0, we'll draw a closed circle (or a filled-in dot) at 0 and shade the number line to the left, indicating all values less than 0. Similarly, since our solution includes values greater than or equal to 12, we'll draw a closed circle at 12 and shade the number line to the right, indicating all values greater than 12.

The unshaded portion of the number line between 0 and 12 represents the values of x that do not satisfy the original inequality. Any number in that range, when plugged into |x-6|, will result in a value less than 6. The shaded portions, on the other hand, represent the values of x that do satisfy the inequality. Choose any number from those shaded regions, plug it into |x-6|, and you'll find that the result is always greater than or equal to 6. This visual representation can be incredibly helpful in understanding the solution and verifying that it makes sense.

Conclusion

Alright, guys! We've successfully solved the absolute value inequality |x-6| ≥ 6 algebraically. We broke it down into two scenarios, solved each scenario separately, and then combined the solutions to get the complete solution set: (-∞, 0] ∪ [12, ∞). We also visualized the solution on a number line to gain a deeper understanding of what it represents. Remember, the key to solving absolute value inequalities is to consider both the positive and negative cases of the expression inside the absolute value. With practice, you'll become a pro at tackling these types of problems! Keep up the great work, and don't be afraid to ask questions if you get stuck. Happy solving!