Solving |4x-6| > 10 A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of inequalities, specifically tackling the absolute value inequality |4x - 6| > 10. This type of problem might seem intimidating at first, but trust me, with a systematic approach, it's totally manageable. We'll break it down step by step, ensuring you grasp the underlying concepts and can confidently solve similar problems in the future. This comprehensive guide aims to provide you with a clear understanding, making those math challenges a little less challenging. Whether you are a student grappling with homework or someone brushing up on your algebra skills, this explanation is tailored to help you master absolute value inequalities.

Understanding Absolute Value Inequalities

Before we jump into solving our specific inequality, let's first understand absolute value inequalities. Absolute value, in simple terms, represents the distance of a number from zero on the number line. It's always non-negative. So, |5| is 5, and |-5| is also 5. When we throw inequalities into the mix, we're essentially asking for the range of values that satisfy a certain distance condition. For instance, |x| < 3 means we're looking for all numbers whose distance from zero is less than 3, which translates to -3 < x < 3. Conversely, |x| > 3 means we want numbers whose distance from zero is greater than 3, leading to two separate intervals: x < -3 or x > 3. This 'or' condition is crucial and forms the basis for solving inequalities like the one we're tackling today.

Understanding this concept of distance is paramount. It helps visualize what the inequality is actually asking. Think of it as a boundary – the absolute value inequality defines a region on the number line that our solutions must fall outside or inside of, depending on the inequality sign. When dealing with expressions inside the absolute value, like our 4x - 6, we're essentially shifting and scaling this region. The key is to correctly interpret how these shifts and scales affect the final solution. So, before we move on, make sure you're comfortable with the basic idea: absolute value measures distance, and inequalities dictate whether we're looking for distances greater than or less than a certain value.

Breaking Down |4x - 6| > 10

Now, let's tackle the inequality |4x - 6| > 10 step-by-step. The absolute value expression |4x - 6| represents the distance of the expression 4x - 6 from zero. The inequality |4x - 6| > 10 tells us that this distance must be greater than 10. This is the crux of the problem, and understanding this interpretation is vital. Because of the absolute value, this inequality actually translates into two separate cases, and we must consider both to find the complete solution. This is a common point where students might stumble, so let's make it crystal clear: we need to account for both when the expression inside the absolute value is positive and when it is negative.

So, what are these two cases? First, we consider the case where 4x - 6 is already greater than 10. This is the straightforward scenario where the expression inside the absolute value is positive. The second case is where 4x - 6 is less than -10. Why -10? Because the absolute value of any number less than -10 will still be greater than 10. For example, |-11| is 11, which is greater than 10. This might seem a bit tricky at first, but it's fundamental to grasping absolute value inequalities. Recognizing these two distinct possibilities is the first major hurdle cleared. From here, we're essentially dealing with two linear inequalities, which are much easier to handle individually.

Case 1: 4x - 6 > 10

Let's solve the first case: 4x - 6 > 10. This inequality represents the scenario where the expression inside the absolute value is positive and its value is greater than 10. To solve it, we're going to use basic algebraic manipulation, just like solving any other linear inequality. The goal is to isolate 'x' on one side of the inequality. We'll start by adding 6 to both sides of the inequality. Remember, whatever we do to one side, we must do to the other to maintain the balance. This gives us 4x > 16. This step is crucial because it simplifies the inequality, bringing us closer to isolating 'x'. Adding the same number to both sides doesn't change the inequality itself, it just rearranges the terms.

Now, to isolate 'x' completely, we need to get rid of the 4 that's multiplying it. We do this by dividing both sides of the inequality by 4. Again, it's vital to perform the same operation on both sides. Dividing by a positive number doesn't flip the inequality sign, which is an important rule to remember. This step gives us x > 4. This is the solution for the first case. It means any value of 'x' greater than 4 will satisfy the condition 4x - 6 > 10. So, values like 5, 6, 7, and so on, are all part of the solution set for this case. Keep in mind that this is only half of the solution, as we still need to consider the second case where 4x - 6 is negative.

Case 2: 4x - 6 < -10

Now, let's tackle the second case: 4x - 6 < -10. This is where things get a little trickier, but stay with me! This case represents the situation where the expression inside the absolute value, 4x - 6, is negative and its magnitude is large enough that its absolute value is still greater than 10. Remember, we're dealing with absolute value, so a negative number far from zero also satisfies the original inequality. Solving this inequality follows the same algebraic principles as the first case, but it's essential to pay attention to the negative sign. We start by adding 6 to both sides, just like before. This gives us 4x < -4. Adding 6 to both sides helps to isolate the term with 'x', making the next step clearer.

Next, we divide both sides by 4 to isolate 'x'. Dividing by a positive number doesn't change the direction of the inequality sign. This crucial step gives us x < -1. This is the solution for the second case. It means any value of 'x' less than -1 will satisfy the condition 4x - 6 < -10. Numbers like -2, -3, -4, and so on, are part of this solution set. Now, we have the solutions for both cases: x > 4 and x < -1. It's important to realize that both these solutions are valid and together they make up the complete solution to the original absolute value inequality. Next, we'll combine these solutions to get the final answer.

Combining the Solutions

Okay, guys, we've solved both cases individually, and now it's time to combine the solutions. This is a crucial step to get the complete picture of what values of 'x' satisfy the inequality |4x - 6| > 10. We found that x > 4 in the first case and x < -1 in the second case. These are two distinct intervals on the number line. The key here is to recognize that these solutions are connected by an