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

by TextBrain Team 58 views

Hey everyone! Today, we're diving into the world of absolute value inequalities. We'll be tackling the inequality |3x - 4| > 2, and I'll walk you through the steps to find its solution set. This might seem a bit tricky at first, but trust me, once you understand the basic principles, you'll be solving these inequalities like a pro. So, grab your pens and paper, and let's get started!

Understanding the Problem: |3x - 4| > 2

First off, let's break down what this inequality means. The expression |3x - 4| represents the absolute value of (3x - 4). Remember, the absolute value of a number is its distance from zero on the number line. This means the absolute value is always non-negative. The inequality |3x - 4| > 2 is asking us to find all the values of 'x' for which the absolute value of (3x - 4) is greater than 2. Think of it this way: we're looking for all the numbers that are more than 2 units away from 0 when we apply the operation inside the absolute value, which is 3x - 4. Because this is the foundation for tackling more complex inequalities involving the absolute value, we must remember the absolute value is about the distance of the number from zero on a number line.

To solve this, we need to consider two separate cases because the expression inside the absolute value bars can be either positive or negative. If (3x - 4) is positive, then |3x - 4| = 3x - 4. If (3x - 4) is negative, then |3x - 4| = -(3x - 4). We will solve each of these conditions separately and then merge the results to find the complete solution set. This is because the definition of absolute value implies two different possibilities, and we must examine both to guarantee our answer considers all possible scenarios. We need to take care to ensure we account for all potential cases, since both positive and negative values may provide us with different solutions to the original inequality. Furthermore, the understanding of absolute values and the associated operations are fundamental in many areas of mathematics, including calculus and advanced algebra. Therefore, a solid grasp of the concepts involved is extremely valuable.

The Key Property: Absolute Value Inequalities

To solve absolute value inequalities like this, we use a fundamental property. It's our secret weapon, the key to unlocking the solution. Here it is:

If |x| > a, then x < -a OR x > a

This property tells us that if the absolute value of 'x' is greater than 'a', then 'x' must either be less than '-a' or greater than 'a'. This is because numbers that are further away from zero on either side of the number line will satisfy the inequality.

Now, let's apply this property to our inequality |3x - 4| > 2. We can see that the number inside the absolute value bars is (3x - 4). We have a number greater than 2, so based on the property listed above we will use this property to solve the inequality. Using the property, we can rewrite our original inequality into two separate inequalities, which is fundamental to understanding how to approach these types of problems. In each of these inequalities, we're isolating a different possibility and working through the math. Remember, these steps are designed to ensure we arrive at the correct solution set, which will include all of the values that make our original statement true. That means the work we do here is all about determining the specific values or ranges of values for x that fulfill the condition imposed in our initial inequality. We will use this knowledge to solve each inequality, getting us closer to the final answer.

Step-by-Step Solution

Now, let's break down the solution step-by-step. We'll use the property we just discussed and solve for 'x' in two different cases.

Case 1: 3x - 4 < -2

In this case, the expression inside the absolute value, (3x - 4), is negative. This directly applies to the left side of the inequality as defined by our key property. We will go through a series of algebraic steps, carefully maintaining the integrity of the inequality at each stage. The goal is to isolate 'x' on one side of the inequality, and in doing so, we’ll be able to pinpoint the range of x values which fulfill the original statement. Each step we take is necessary to move us toward our final solution. Solving this particular case helps define the possible x values, and finding these values requires the systematic application of arithmetic rules and inequalities.

  1. Add 4 to both sides: 3x - 4 + 4 < -2 + 4 3x < 2

  2. Divide both sides by 3: 3x / 3 < 2 / 3 x < 2/3

So, in this case, x must be less than 2/3.

Case 2: 3x - 4 > 2

Here, the expression inside the absolute value, (3x - 4), is positive. This aligns with the right side of the inequality as outlined by our fundamental property. As in Case 1, we'll follow a precise sequence of operations, each designed to guide us closer to the final solution. Each mathematical operation will ensure we uphold the rules of algebra while we isolate x. By following a well-defined series of steps, we ensure that our process is free of ambiguity, making it easier to catch any potential errors along the way. Carefully following these steps will provide us with another set of possible values for x that satisfy the original inequality.

  1. Add 4 to both sides: 3x - 4 + 4 > 2 + 4 3x > 6

  2. Divide both sides by 3: 3x / 3 > 6 / 3 x > 2

So, in this case, x must be greater than 2.

The Solution Set

Now we have the solution for both cases: x < 2/3 OR x > 2. That means any value of 'x' that is less than 2/3 or greater than 2 will satisfy the original inequality |3x - 4| > 2. We can visualize this solution on a number line. You'll see an open circle at 2/3 and an arrow pointing to the left (indicating all values less than 2/3). You'll also see an open circle at 2 and an arrow pointing to the right (indicating all values greater than 2). These are the x values that we can plug into our original equation to find it correct. Remember, open circles mean that these values are not included in the solution set. This visualization can greatly assist in understanding how the solution is structured, where the values lie, and what parts of the number line are included in our answer.

Therefore, the solution set for the inequality |3x - 4| > 2 is x < 2/3 or x > 2.

Conclusion

And there you have it! We've successfully solved the absolute value inequality |3x - 4| > 2. We broke down the problem into two manageable cases, applied the key property of absolute value inequalities, and found the solution set. Remember, practice makes perfect! The more you practice these types of problems, the more comfortable you'll become with the concepts.

Key Takeaways:

  • Understand the properties of absolute values.
  • Break down the problem into two cases.
  • Apply the key property: |x| > a -> x < -a or x > a
  • Solve for 'x' in each case.
  • Combine the results to get the solution set.

Keep practicing, and you'll be mastering absolute value inequalities in no time. Thanks for joining me today, and I'll catch you in the next lesson! Don't hesitate to ask if you have any more questions, and happy learning, everyone!