Solving Absolute Value Inequalities: A Step-by-Step Guide

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Solving Absolute Value Inequalities: A Step-by-Step Guide

Let's break down how to solve the absolute value inequality 3|x-6|-8 > 16. Absolute value problems can seem tricky, but with a systematic approach, they become quite manageable. We'll go through each step in detail, so you can tackle similar problems with confidence. Understanding absolute value is crucial for many areas of mathematics, and mastering these inequalities is a valuable skill.

1. Isolate the Absolute Value Term

First things first, we need to get the absolute value term all by itself on one side of the inequality. Our starting inequality is:

3|x-6| - 8 > 16

To isolate the absolute value, we'll add 8 to both sides. Think of it like solving a regular equation – we want to undo the operations that are messing with our absolute value.

3|x-6| - 8 + 8 > 16 + 8

This simplifies to:

3|x-6| > 24

Now, we need to get rid of that pesky 3 that's multiplying the absolute value. To do this, we'll divide both sides by 3:

(3|x-6|) / 3 > 24 / 3

Which gives us:

|x-6| > 8

Great! Now we have the absolute value term isolated. This is a critical step because it allows us to move on to the next phase of solving the inequality. Remember, the goal here is to get the absolute value expression by itself so we can properly interpret what the inequality is telling us.

2. Understanding Absolute Value

Before we proceed, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. For example, |3| = 3 and |-3| = 3. Absolute value always returns a non-negative value.

When we have an inequality like |x-6| > 8, it means that the distance between 'x-6' and zero is greater than 8. This leads to two separate cases that we need to consider. This is perhaps the most important thing to remember. A single absolute value inequality will always lead to two separate linear inequalities that you will need to solve.

3. Split into Two Cases

Because of the nature of absolute value, we need to consider two separate cases:

  • Case 1: The expression inside the absolute value is positive or zero. In this case, we can simply remove the absolute value bars.
  • Case 2: The expression inside the absolute value is negative. In this case, we remove the absolute value bars and multiply the expression by -1 (which is the same as changing its sign).

So, |x-6| > 8 becomes two separate inequalities:

  • Case 1: x - 6 > 8
  • Case 2: x - 6 < -8 (Notice that the inequality sign flips when we consider the negative case!)

It's essential to understand why we flip the inequality sign in Case 2. When we're dealing with negative numbers, the number further away from zero has a smaller value. For instance, -9 is less than -8, even though 9 is greater than 8. This is a common area where people make mistakes, so pay close attention here!

4. Solve Each Inequality

Now that we have our two cases, we simply solve each inequality separately.

Case 1: x - 6 > 8

To solve for x, we add 6 to both sides:

x - 6 + 6 > 8 + 6

x > 14

So, in the first case, x must be greater than 14.

Case 2: x - 6 < -8

Similarly, we add 6 to both sides:

x - 6 + 6 < -8 + 6

x < -2

In the second case, x must be less than -2.

5. Write the Solution Set

Now that we've solved both cases, we need to write the solution set. The solution set includes all values of x that satisfy either inequality. In this case, x can be greater than 14 OR less than -2. In interval notation, we write this as:

(-∞, -2) βˆͺ (14, ∞)

The symbol 'βˆͺ' means 'union,' which indicates that we're combining the two intervals into one solution set.

6. Verification and Testing

It's always a good idea to check our work. We can do this by picking a value from each interval in our solution set and plugging it back into the original inequality to see if it holds true.

  • Test x = -3 (from the interval (-∞, -2))

3|(-3) - 6| - 8 > 16

3|-9| - 8 > 16

3(9) - 8 > 16

27 - 8 > 16

19 > 16 (This is true!)

  • Test x = 15 (from the interval (14, ∞))

3|(15) - 6| - 8 > 16

3|9| - 8 > 16

3(9) - 8 > 16

27 - 8 > 16

19 > 16 (This is also true!)

Since the inequality holds true for values from both intervals, our solution set is likely correct.

Common Mistakes to Avoid

  • Forgetting to Flip the Inequality Sign: This is a very common mistake. Remember that when you're dealing with the negative case of the absolute value, you need to flip the inequality sign.
  • Not Isolating the Absolute Value: You must isolate the absolute value term before splitting the inequality into two cases. Otherwise, you'll end up with incorrect solutions.
  • Incorrectly Combining the Intervals: Make sure you understand when to use 'and' versus 'or' when combining the solutions from the two cases. In this problem, we use 'or' because x can be in either interval.
  • Arithmetic Errors: Always double-check your arithmetic, especially when adding or subtracting negative numbers.

Practice Problems

To solidify your understanding, try solving these similar inequalities:

  1. 2|x + 3| - 5 > 7
  2. 4|2x - 1| + 3 > 11
  3. |x/2 + 1| - 4 > 0

By working through these problems, you'll build your confidence and become more comfortable with solving absolute value inequalities.

Conclusion

Solving absolute value inequalities involves isolating the absolute value term, splitting the problem into two cases, solving each case separately, and then combining the solutions into a solution set. Remember to flip the inequality sign in the negative case and always verify your solution. With practice, you'll become proficient at solving these types of problems. So go ahead and tackle those inequalities with confidence, guys! You've got this! Understanding these concepts is essential for future math endeavors!