Solving Inequalities In Real Numbers: A Step-by-Step Guide

Hey guys! Today, we're going to dive into the world of solving inequalities in real numbers. This can seem a little tricky at first, but trust me, with a clear understanding of the steps involved, you'll be acing these problems in no time. We'll be working through the inequality: (3x-1)/4 + (2x-5)/3 < (7x-11)/12 + 1 * 1/2. Let's break it down into manageable chunks.

Understanding the Basics of Inequalities

Before we jump into the problem, let's make sure we're all on the same page with the fundamentals. Inequalities, you know, are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The goal when solving an inequality is pretty similar to solving an equation: we want to isolate the variable (in our case, 'x') on one side of the inequality sign. But here's the kicker: there's one important rule to remember, which we'll get to soon. This entire process revolves around maintaining the balance of the inequality. Whatever operation you perform on one side of the inequality, you must perform on the other side as well. This is the cornerstone of solving inequalities correctly. Think of it like a seesaw – to keep it balanced, you need to add or remove the same weight from both sides. Failing to do so throws everything off, leading to incorrect solutions. Also, remember that we're dealing with real numbers, so any value of 'x' that satisfies the inequality belongs to the solution set. Understanding the implications of the inequality signs, especially the direction they point, is crucial for interpreting your final answer. Mastering these foundational concepts will make navigating complex inequalities a breeze.

Now, let's consider the differences between solving equations and inequalities. In equations, we're typically looking for a specific value (or values) of the variable that makes the equation true. For inequalities, however, we are looking for a range of values. This range is what defines our solution set. For example, if we solve and get x > 2, that means any number greater than 2 is a solution to the inequality. This difference has significant implications for how we solve and represent our solutions.

In essence, solving inequalities helps us determine the range of values that will keep the statement true. This contrasts with finding an exact value like in equations. This difference also affects how we check our answers. In equations, plugging the solution back in is enough. In inequalities, we can check by picking any number that is in the range. If that number satisfies the inequality, that gives us the green light that our answer is probably correct. Understanding these basics is essential to tackle the problem we have at hand.

Step-by-Step Solution

Alright, let's get down to business and solve the inequality: (3x-1)/4 + (2x-5)/3 < (7x-11)/12 + 1 * 1/2. We'll work through it step by step, so follow along! The first thing we need to do is to get rid of the fractions, and a great way to do that is to find the least common multiple (LCM) of the denominators (4, 3, and 12). The LCM of 4, 3, and 12 is 12. Multiply every term in the inequality by 12.

• 12 * [(3x-1)/4] + 12 * [(2x-5)/3] < 12 * [(7x-11)/12] + 12 * (1/2)

Now, simplify each term:

• 3 * (3x-1) + 4 * (2x-5) < 1 * (7x-11) + 6

Next, expand the terms by distributing:

• 9x - 3 + 8x - 20 < 7x - 11 + 6

Combine like terms on both sides:

• 17x - 23 < 7x - 5

Now, we'll isolate the 'x' terms on one side and the constants on the other. Subtract 7x from both sides:

• 17x - 7x - 23 < 7x - 7x - 5

• 10x - 23 < -5

Add 23 to both sides:

• 10x - 23 + 23 < -5 + 23

• 10x < 18

Finally, divide both sides by 10 to solve for 'x':

• x < 18/10

• x < 9/5

So, the solution to the inequality is x < 9/5. That's it! Now, let's talk about what this answer means.

Interpreting the Solution and Checking Your Work

So, what does x < 9/5 actually tell us, and how can we be sure we've got the right answer, guys? The solution x < 9/5 means that any real number less than 9/5 (or 1.8) will satisfy the original inequality. Pretty cool, right? But wait, how do we check our work to make sure we're on the right track? Well, there are a couple of ways.

First, you can test a value. Pick a number that is less than 9/5 (but not equal to it). For instance, let's try x = 1. Plug this value back into the original inequality and see if it holds true: (3(1)-1)/4 + (2(1)-5)/3 < (7(1)-11)/12 + 1/2. That simplifies to (2/4) + (-3/3) < (-4/12) + (1/2), which becomes (1/2) - 1 < (-1/3) + (1/2). Further simplifying gives us -1/2 < 1/6. This is true! Because the inequality holds when x = 1, this gives us good reason to believe we're on the right track. Remember though, testing one number doesn't fully prove it, but it's a good way to see if we're on the right track.

Second, you can visualize the solution on a number line. Draw a number line and mark 9/5 (1.8) on it. Since our solution is x < 9/5, you'd shade the number line to the left of 9/5, indicating that all the values in that direction satisfy the inequality. You would use an open circle at 9/5 to show that 9/5 is not included in the solution set. This visualization provides a clear picture of the range of solutions.

Finally, always double-check your calculations. It's easy to make a small arithmetic error, especially when you're dealing with fractions or negative numbers. Go back through your steps and make sure you haven't made any mistakes in simplifying or distributing. Remember, attention to detail is your friend when it comes to solving inequalities. So, when in doubt, rework it!

Common Mistakes and How to Avoid Them

Alright, we've walked through the solution, but it's equally important to know the common pitfalls. The biggest mistake students make when solving inequalities involves multiplying or dividing both sides by a negative number. This is where the golden rule comes in: when you multiply or divide by a negative number, you must flip the inequality sign. For example, if you have -2x > 4, you'll need to divide both sides by -2, and the inequality becomes x < -2. A lot of students forget this, and it leads to incorrect answers. Be super mindful of this rule; it's the key to getting these problems right!

Another common mistake is carelessness with signs. Be very careful when you're distributing and when you're combining like terms. A misplaced negative sign can completely change your answer. Take your time, and double-check each step. Don't rush; it's better to go slow and be accurate than to rush and make a mistake. Also, don't get tripped up by fractions! Take your time, and make sure you’re applying the correct operations (like finding the LCM) to handle them correctly.

Finally, remember to simplify your answers fully. Leaving your answer as an improper fraction is okay, but always reduce the fraction to its lowest terms. Also, if the problem requires it, express your solution in interval notation (e.g., (-∞, 9/5)). Doing so shows you have a complete understanding of the solution set. Avoid these mistakes, and you'll be well on your way to acing inequality problems! Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become, and the fewer mistakes you'll make.

Conclusion: Mastering Inequalities

There you have it, guys! We've successfully solved an inequality and now you're well-equipped to tackle similar problems. Remember, the core of solving inequalities lies in a few key principles: isolating the variable, remembering to flip the inequality sign when multiplying or dividing by a negative number, and carefully checking your work. Take your time, break down each problem into smaller, manageable steps, and always double-check your calculations. Practice is key, so don't be afraid to work through plenty of examples to build your confidence and solid understanding. Keep up the great work, and you'll become a pro at solving inequalities in no time! Keep practicing, and don't be afraid to ask for help when you need it.

Good luck, and happy solving!