Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of inequalities. Specifically, we're going to solve the inequality: -2(x + 3) < 14. Don't worry, it might look a bit intimidating at first, but trust me, it's totally manageable. We'll break it down step by step, so you can easily understand how to find the value of x that satisfies this inequality. Remember, inequalities are super useful in real-life scenarios, from budgeting to understanding speed limits. So, let's get started and make solving inequalities a piece of cake!

Understanding the Basics of Inequalities

Alright, before we jump into the problem, let's get our foundations straight. An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have a single solution, inequalities usually have a range of solutions. Think of it like this: an equation is like finding a specific address, while an inequality is like defining a neighborhood. Our goal is to find the values of x that make the left side of the inequality smaller than 14.

Now, the key here is to remember that solving inequalities is very similar to solving equations. We'll use the same algebraic principles: adding, subtracting, multiplying, and dividing to isolate the variable x. However, there's one important rule to keep in mind, and we'll get to that in a bit. Just like with equations, our objective is to isolate x on one side of the inequality sign. We want to get x by itself so we can see what range of values it can take on. This process involves a series of logical steps, each designed to simplify the inequality while maintaining its truth. We'll work carefully to ensure that we don't accidentally change the inequality's direction, because that would completely change our answer! So, let's get cracking with our specific problem, and you'll see how easy it is to handle these types of questions. Keep in mind that practice is key, so the more problems you try, the more comfortable you'll become. By the end of this guide, you’ll be solving inequalities like a pro, and be ready to tackle more complex problems with confidence.

Step-by-Step Solution to -2(x + 3) < 14

Alright, buckle up, because here comes the fun part! We're going to solve -2(x + 3) < 14 step-by-step. I'll walk you through each move, explaining why we're doing it, so you'll not only get the answer but also understand the how and why behind it.

Step 1: Distribute

The first thing we need to do is get rid of those parentheses. To do that, we'll distribute the -2 across both terms inside the parentheses. This means multiplying -2 by x and -2 by 3.

So, -2 * x becomes -2x, and -2 * 3 becomes -6. Our inequality now looks like this: -2x - 6 < 14.

See? We're already making progress! This step simplifies the expression, making it easier to isolate x. Always remember to distribute carefully, paying attention to the signs. This is a common place where mistakes can happen, so take your time and double-check your work. We are essentially rewriting the left side of the inequality in a way that makes it easier to work with. Think of it as unfolding a complicated origami shape into simpler folds; the basic form remains the same, but the structure is easier to manipulate.

Step 2: Isolate the Variable Term

Now, we need to get the term with x by itself on one side of the inequality. To do this, we'll get rid of the -6. Since we're subtracting 6, we'll do the opposite and add 6 to both sides of the inequality. This is crucial: whatever you do to one side, you must do to the other to keep the inequality balanced.

So, -2x - 6 + 6 < 14 + 6. This simplifies to -2x < 20. Adding 6 to both sides cancels out the -6 on the left side, leaving us with just the x term. The right side becomes 20, as 14 + 6 = 20. Adding the same value to both sides is like adding equal weights to each side of a balance scale. The scale remains balanced, and our inequality remains true. It's all about maintaining that balance to avoid changing the solution. If we didn't add the same value to both sides, the inequality would no longer hold, and our final answer would be wrong. This step sets us up to isolate x in the next step.

Step 3: Solve for x

We're in the home stretch now! We have -2x < 20. To isolate x, we need to get rid of the -2 that's multiplying it. We'll do this by dividing both sides of the inequality by -2.

Here's the super important part: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is a critical rule to remember!

So, -2x / -2 > 20 / -2. Notice how the < sign changed to a > sign. This gives us x > -10.

Why do we flip the sign? Well, when you multiply or divide by a negative number, you're essentially changing the order of the numbers on the number line. For example, consider -1 and -2. -1 is greater than -2. But if you multiply both by -1, you get 1 and 2, and 1 is less than 2. This is why we flip the sign. It ensures that the inequality remains true. In this case, we're saying that x is greater than -10. Any number larger than -10 will satisfy the original inequality. For instance, if you plug in 0 for x into the original inequality, you'll see that it works because 0 is greater than -10. Congratulations, you've solved the inequality!

Understanding the Solution: x > -10

So, what does x > -10 actually mean? Well, it means that x can be any number that is greater than -10. It’s not just one number, but an infinite set of numbers! On a number line, this would be all the numbers to the right of -10, not including -10 itself. We use an open circle at -10 to show that -10 is not included in the solution.

Think about it this way: any number bigger than -10, like -9, -5, 0, 5, 10, and so on, will make the original inequality true. If you substitute any of these numbers back into the original inequality -2(x + 3) < 14, you'll find that the left side is indeed less than 14.

This is a range of possible solutions. Unlike an equation, which gives you a single value for x, an inequality tells you the possible values for x. Understanding this range is crucial for applying inequalities to real-world problems. Whether you're budgeting, figuring out distances, or interpreting scientific data, inequalities are your friends. The ability to correctly interpret and understand the solution is just as important as the solving itself, because it is how we take the math out of the theoretical realm and into the practical one.

Checking Your Answer

Always, always check your answer! It's super easy to make a small mistake along the way, so let's make sure we got this right. A quick way to check your solution is to pick a number that's greater than -10 (because our solution is x > -10) and plug it back into the original inequality.

Let's choose 0 (it's easy to work with!). Substitute x = 0 into -2(x + 3) < 14:

-2(0 + 3) < 14 -2(3) < 14 -6 < 14

Yep! -6 is less than 14. This means our answer, x > -10, is likely correct.

You can also try another number, like -5. Substituting x = -5:

-2(-5 + 3) < 14 -2(-2) < 14 4 < 14

Again, it works! And just to be extra sure, let's try a number less than -10, like -11. Substituting x = -11:

-2(-11 + 3) < 14 -2(-8) < 14 16 < 14

Nope! 16 is not less than 14. This confirms that our solution, x > -10, is correct.

Checking your answer is a fantastic habit. It helps you catch any errors you might have made during the solving process and builds your confidence. Plus, it solidifies your understanding of inequalities. Think of it as a quality check; it ensures your work meets the standard and gives you peace of mind.

Tips and Tricks for Solving Inequalities

Alright, you've conquered your first inequality! Let's level up your skills with some handy tips and tricks.

• Practice, Practice, Practice: The more problems you solve, the better you'll get. Work through various examples to build your confidence and fluency. Use different textbooks, online resources, or practice problems provided by your teacher to get your practice in. Doing a variety of problems is the best way to develop intuition and to master solving these problems, so don't be afraid to try problems that seem difficult.
• Show Your Work: Always write down each step. This helps you avoid careless mistakes and makes it easier to find and correct any errors. Showing your work is a critical habit to develop in mathematics. Not only does it help you to be more accurate, it's also invaluable when reviewing for tests, and when asking for help from teachers or tutors. Clear, well-organized steps make it easy to follow your logic.
• Double-Check the Sign: Remember to flip the inequality sign when multiplying or dividing by a negative number. This is a common point of error, so pay close attention.
• Use a Number Line: Visualizing the solution on a number line can help you understand the range of values that satisfy the inequality. It makes the abstract concept more concrete.
• Break It Down: If the inequality looks complicated, break it down into smaller, more manageable steps. This reduces the chance of making errors and makes the problem less daunting.
• Know the Rules: Remember the basic rules of algebra. Addition, subtraction, multiplication, and division need to be applied to both sides of the inequality to maintain balance. The sign change for multiplication/division with negatives is your most important rule to remember.

Conclusion: You've Got This!

Congratulations, guys! You've successfully navigated the world of inequalities and solved -2(x + 3) < 14. You've learned how to isolate the variable, distribute, and most importantly, how to flip the inequality sign when necessary. Remember, the key to mastering inequalities (and any math problem) is practice and understanding. Keep practicing, keep learning, and don't be afraid to ask for help when you need it. You are now equipped with the fundamental knowledge and skills to tackle inequalities with confidence. Every step is an achievement; every problem solved adds to your understanding.

So go forth, conquer those inequalities, and keep up the amazing work! You've got this, and with consistent effort, you'll be acing those math problems in no time. Keep the positive attitude and focus, and you'll be well on your way to math mastery! Until next time, keep those math muscles flexing! Have a great day!