Hey guys! Let's dive into the fascinating world of inequalities, specifically those involving absolute values. If you've ever felt a little intimidated by these problems, don't worry! We're going to break it down step by step, making it super clear and easy to understand. Absolute value inequalities might seem tricky at first, but with a systematic approach and a bit of practice, you’ll be solving them like a pro in no time. Absolute value represents the distance of a number from zero on the number line, regardless of direction. So, whether the number is positive or negative, its absolute value is always non-negative. This concept is crucial when dealing with inequalities, as it introduces two possibilities that we need to consider. When we see an absolute value expression within an inequality, we're essentially dealing with a range of values rather than a single value. This is because the expression inside the absolute value can be either positive or negative, and we need to account for both scenarios. Understanding this dual nature is the key to correctly solving these types of problems. Let’s start with a basic example to illustrate this point. Consider the absolute value inequality |x| < 3. This inequality is asking us to find all the values of x that are less than 3 units away from zero. This includes both positive and negative values. The solution set will therefore include numbers between -3 and 3, excluding the endpoints. This is because any number within this range, when its distance from zero is calculated, will be less than 3. On the other hand, if we had |x| > 3, we would be looking for all the values of x that are more than 3 units away from zero. This means x could be greater than 3 or less than -3. This gives us two distinct intervals for the solution set. In summary, when dealing with absolute value inequalities, it's crucial to remember that the absolute value function introduces two cases: one where the expression inside the absolute value is positive or zero, and another where it is negative. We need to address both of these cases to find the complete solution to the inequality. Now, let’s move on to a specific example and see how this works in practice.
Deconstructing the Inequality: |(1/2)x - 2| < 3/2
Our mission, should we choose to accept it (and we do!), is to solve the inequality |(1/2)x - 2| < 3/2. This might look a tad intimidating, but trust me, it's totally manageable. We're going to take it apart piece by piece, just like solving a puzzle. This inequality involves an absolute value expression, so the first thing we need to recognize is that we’re essentially dealing with two separate inequalities wrapped into one. The expression inside the absolute value, (1/2)x - 2, can be either positive or negative, and we need to consider both possibilities. When we think about absolute values, remember that |a| < b translates to two inequalities: -b < a < b. This is because the distance from 'a' to zero must be less than 'b', which means 'a' can be any number between -b and b. Applying this concept to our inequality, |(1/2)x - 2| < 3/2, we can rewrite it as two separate inequalities: -3/2 < (1/2)x - 2 < 3/2. This combined inequality now gives us a range within which the expression (1/2)x - 2 must fall. To solve this, we'll isolate the variable x by performing the same operations on all parts of the inequality. This ensures that the inequality remains balanced and that we don't inadvertently change the solution set. The first step is to eliminate the constant term, -2, from the middle part of the inequality. We can do this by adding 2 to all three parts: -3/2 + 2 < (1/2)x - 2 + 2 < 3/2 + 2. Simplifying this gives us: 1/2 < (1/2)x < 7/2. Now we need to get rid of the coefficient (1/2) multiplying x. We can do this by multiplying all parts of the inequality by 2: 2 * (1/2) < 2 * (1/2)x < 2 * (7/2). This simplifies to: 1 < x < 7. So, we've found that x must be greater than 1 and less than 7. This gives us the solution set for our original inequality. But what does this solution actually mean? It means that any value of x between 1 and 7, when plugged into the original inequality, will make the inequality true. This range represents all the numbers that satisfy the condition of being less than 3/2 units away from 2 when multiplied by 1/2. In the next section, we’ll delve deeper into how to represent this solution and verify our answer.
Step-by-Step Solution: Cracking the Code
Okay, let's get our hands dirty and solve this thing step by step. We're going to take those initial inequalities and massage them until we have 'x' all by its lonesome. First, remember our rewritten inequality: -3/2 < (1/2)x - 2 < 3/2. The goal here is to isolate 'x' in the middle. To do this, we need to get rid of the -2 and the 1/2 that are hanging out with our 'x'. The golden rule of inequalities (and equations) is that whatever you do to one part, you gotta do to all parts. So, let's start by adding 2 to all three sections of the inequality. This will cancel out the -2 in the middle. Adding 2 to all parts, we get: -3/2 + 2 < (1/2)x - 2 + 2 < 3/2 + 2. Now, let's simplify those fractions. Remember, 2 is the same as 4/2, so we have: -3/2 + 4/2 < (1/2)x < 3/2 + 4/2. This simplifies to: 1/2 < (1/2)x < 7/2. Great! We've gotten rid of the -2. Now, we need to deal with that 1/2 that's multiplying our 'x'. The opposite of multiplying is dividing, but since we're dealing with fractions, it's easier to think about multiplying by the reciprocal. The reciprocal of 1/2 is 2, so we'll multiply all parts of the inequality by 2. Multiplying by 2, we get: 2 * (1/2) < 2 * (1/2)x < 2 * (7/2). Simplifying this, we have: 1 < x < 7. Voila! We've solved for 'x'. This tells us that 'x' can be any number between 1 and 7, but it can't be 1 or 7 themselves (because the inequality signs are strictly less than, not less than or equal to). This solution is a range of values, which is typical for inequalities. Now, let's think about what this solution means in the context of our original problem. It means that if we pick any number between 1 and 7, plug it into the original absolute value inequality, and simplify, the inequality will hold true. For instance, if we pick x = 4 (which is between 1 and 7), we can plug it into the original inequality to check. But before we do that, let's talk about how to represent this solution graphically and in interval notation.
Representing the Solution: Intervals and Number Lines
So, we've cracked the code and found that 1 < x < 7. Awesome! But how do we show this solution in a way that's crystal clear? There are two main ways: using a number line and using interval notation. Let's start with the number line. A number line is a visual way to represent our solution. Imagine a line stretching out infinitely in both directions, with zero in the middle and numbers increasing to the right and decreasing to the left. Our solution, 1 < x < 7, means we want to highlight all the numbers between 1 and 7, but not including 1 and 7 themselves. To do this on the number line, we'll draw a line segment between 1 and 7. Since we're not including 1 and 7, we'll use open circles (also sometimes called parentheses) at these points. An open circle indicates that the endpoint is not part of the solution. If we were including the endpoints (i.e., if our inequality was 1 ≤ x ≤ 7), we'd use closed circles (or square brackets) instead. The line segment between the open circles represents all the numbers that satisfy our inequality. It's a clear, visual way to see the range of solutions. Now, let's talk about interval notation. Interval notation is a shorthand way to write the solution set using parentheses and brackets. It's a neat and concise way to express the range of values. For our solution, 1 < x < 7, we'll use parentheses because we're not including the endpoints. The interval notation for this solution is (1, 7). The parentheses tell us that 1 and 7 are not included in the solution set. If we were including the endpoints, we'd use square brackets instead. For example, if our solution was 1 ≤ x ≤ 7, the interval notation would be [1, 7]. The square brackets indicate that 1 and 7 are part of the solution. Interval notation is read from left to right, just like the number line. The first number is the lower bound of the interval, and the second number is the upper bound. If the interval extends infinitely in either direction, we use the infinity symbol (∞) or negative infinity symbol (-∞). For example, if our solution was x > 5, the interval notation would be (5, ∞). If our solution was x ≤ -2, the interval notation would be (-∞, -2]. Now that we know how to represent our solution, let's take a moment to verify our answer and make sure we haven't made any mistakes.
Verification: Does Our Solution Hold Up?
Time to play detective and make sure our solution actually works! We found that 1 < x < 7, but let's put that to the test. The best way to verify our solution is to pick a few numbers within our range, plug them back into the original inequality, and see if they make it true. It's also a good idea to test numbers outside our range to see that they make the inequality false. This helps confirm that we've found the correct boundaries for our solution. Let's start by picking a number inside our solution range. A nice, easy number between 1 and 7 is 4. So, let's plug x = 4 into our original inequality: |(1/2)x - 2| < 3/2. Substituting x = 4, we get: |(1/2)(4) - 2| < 3/2. Simplifying inside the absolute value: |2 - 2| < 3/2, which becomes |0| < 3/2. Since |0| = 0, we have 0 < 3/2, which is absolutely true! This gives us confidence that our solution is on the right track. Now, let's try a number outside our solution range. A good choice would be x = 0, which is less than 1. Plugging x = 0 into our original inequality, we get: |(1/2)(0) - 2| < 3/2. Simplifying: |-2| < 3/2, which becomes 2 < 3/2. This is false! 2 is not less than 3/2 (which is 1.5). This confirms that numbers outside our range do not satisfy the inequality. Let's try another number outside our range, this time one greater than 7. Let's pick x = 8. Plugging x = 8 into our original inequality, we get: |(1/2)(8) - 2| < 3/2. Simplifying: |4 - 2| < 3/2, which becomes |2| < 3/2. This simplifies to 2 < 3/2, which, as we saw before, is false. So, we've tested numbers inside and outside our solution range, and our results confirm that 1 < x < 7 is indeed the correct solution. This verification process is crucial because it helps us catch any mistakes we might have made during the solving process. It's always a good idea to double-check your work, especially with inequalities, as the direction of the inequality sign can be tricky. By verifying our solution, we can be confident that we've solved the problem correctly. Now, to wrap things up, let's recap the steps we took and highlight the key concepts we learned.
Wrapping Up: Key Takeaways for Absolute Value Inequalities
Alright, guys, we've reached the end of our journey through the world of absolute value inequalities! We've taken a potentially daunting problem and broken it down into manageable steps. Let's recap the key takeaways so you can tackle these problems with confidence. First and foremost, remember the golden rule of absolute values: |a| < b means -b < a < b. This is the foundation for solving inequalities involving absolute values. Whenever you see an absolute value expression within an inequality, immediately think about splitting it into two cases. This is because the expression inside the absolute value can be either positive or negative, and we need to account for both scenarios. In our example, |(1/2)x - 2| < 3/2, we rewrote it as -3/2 < (1/2)x - 2 < 3/2. This step is crucial for correctly solving the inequality. Once you've split the absolute value inequality into its component inequalities, the next step is to isolate the variable. This usually involves performing the same operations (addition, subtraction, multiplication, division) on all parts of the inequality. Remember to keep the inequality balanced – whatever you do to one part, you must do to all parts. In our example, we added 2 to all parts of the inequality and then multiplied all parts by 2 to isolate 'x'. After isolating the variable, you'll have your solution. It's important to represent your solution clearly. We discussed two main ways to do this: using a number line and using interval notation. A number line provides a visual representation of the solution, while interval notation is a concise way to write the solution set using parentheses and brackets. Finally, and this is super important, always verify your solution! Pick numbers within and outside your solution range and plug them back into the original inequality. This will help you catch any mistakes and ensure that your solution is correct. Solving absolute value inequalities might seem challenging at first, but with practice and a systematic approach, you'll become a pro in no time. Remember to break down the problem, apply the golden rule of absolute values, isolate the variable, represent your solution clearly, and always verify your answer. Now go forth and conquer those inequalities!