Solve (X+5)(X-1) < 7: A Step-by-Step Guide
Introduction: Unlocking the Secrets of Polynomial Inequalities
Hey guys! Have you ever stumbled upon a mathematical puzzle that seems a bit daunting at first glance? Well, today, we're going to tackle one of those head-scratchers together: polynomial inequalities. Specifically, we're diving deep into the inequality (X+5)(X-1) < 7. Don't worry if that looks like a jumble of symbols right now. By the end of this guide, you'll not only understand how to solve it but also appreciate the logic behind each step. We'll break down the process into easy-to-follow sections, ensuring you grasp every concept along the way. So, grab your thinking caps, and let's get started!
Why Polynomial Inequalities Matter
Before we jump into the nitty-gritty, let's take a moment to understand why polynomial inequalities are important. You might be wondering, “When will I ever use this in real life?” The truth is, these types of problems pop up in various fields, from engineering and physics to economics and computer science. They help us model situations where quantities aren't just equal but fall within a certain range. For instance, imagine you're designing a bridge and need to ensure the load stays within safe limits, or you're forecasting sales and need to predict when profits will exceed a certain threshold. Polynomial inequalities are your trusty tools in these scenarios. Understanding them not only boosts your math skills but also gives you a powerful problem-solving edge in the real world. So, let’s dive into the exciting realm of polynomial inequalities and see how we can master them together!
What We'll Cover in This Guide
In this comprehensive guide, we'll walk through each stage of solving the polynomial inequality (X+5)(X-1) < 7. We will start by simplifying the inequality, which involves expanding the product and rearranging terms to get a more manageable form. Next, we'll find the critical points, the values of X where the expression equals 7, which are crucial for determining the intervals where the inequality holds true. Then, we'll use these critical points to divide the number line into intervals and test each interval to see if it satisfies the inequality. Finally, we'll put it all together to write the solution in interval notation, giving you a clear and concise answer. We will also touch on the common mistakes to avoid and provide additional tips to ensure your success in solving polynomial inequalities. By the end of this guide, you'll have a solid understanding of how to tackle these problems with confidence. So, let's embark on this mathematical journey and unlock the secrets of polynomial inequalities together!
Step 1: Simplifying the Inequality
Okay, let's get down to business! Our first step in solving the polynomial inequality (X+5)(X-1) < 7 is to simplify it. This means we need to get rid of those parentheses and rearrange the terms so the inequality is easier to work with. Think of it like decluttering your desk before starting a big project – a clean setup makes everything smoother. We'll start by expanding the left side of the inequality. Remember the FOIL method? (First, Outer, Inner, Last) This will help us multiply the two binomials correctly. So, we'll multiply the First terms (X and X), then the Outer terms (X and -1), then the Inner terms (5 and X), and finally the Last terms (5 and -1). Doing this carefully is super important to avoid mistakes. Once we've expanded the expression, we'll combine like terms to simplify it further. This will give us a quadratic expression, which is a polynomial of degree two. After that, we'll need to move all terms to one side of the inequality, leaving zero on the other side. This step is crucial because it sets us up to find the critical points in the next stage. So, let's roll up our sleeves and get simplifying! Remember, accuracy and attention to detail are key here, so take your time and double-check your work.
Expanding the Product (X+5)(X-1)
Alright, let's dive into expanding the product (X+5)(X-1). As we discussed, we'll be using the FOIL method to ensure we multiply every term correctly. Remember, FOIL stands for First, Outer, Inner, Last, and it's a fantastic way to organize our multiplication. First, we multiply the First terms: X * X = X². Easy peasy, right? Next up are the Outer terms: X * -1 = -X. Now, let's move on to the Inner terms: 5 * X = 5X. And finally, the Last terms: 5 * -1 = -5. So, when we put it all together, we have X² - X + 5X - 5. But we're not done yet! We still need to combine those like terms to simplify our expression further. Notice that we have both a -X term and a 5X term. These are like terms because they both contain the variable X raised to the power of 1. So, we can combine them by adding their coefficients: -1 + 5 = 4. This means -X + 5X simplifies to 4X. Now, let's put everything together. Our expanded and simplified expression is X² + 4X - 5. This is a quadratic expression, and it's much easier to work with than the original product. We've successfully expanded the product and simplified the expression, and this is a major step forward in solving our inequality. So, give yourself a pat on the back – you're doing great!
Rearranging the Inequality
Now that we've expanded the left side of the inequality (X+5)(X-1) < 7 to X² + 4X - 5, we need to take the next step and rearrange the entire inequality. The goal here is to get all the terms on one side and leave zero on the other side. This is a crucial step because it allows us to analyze the quadratic expression more easily and find the critical points. To do this, we'll subtract 7 from both sides of the inequality. Remember, whatever we do to one side of an inequality, we must do to the other side to maintain the balance. So, let's take our expression X² + 4X - 5 < 7 and subtract 7 from both sides. This gives us X² + 4X - 5 - 7 < 7 - 7. On the right side, 7 - 7 is, of course, 0. On the left side, we need to combine the constant terms -5 and -7. Adding these together gives us -12. So, our rearranged inequality is X² + 4X - 12 < 0. This is a quadratic inequality, and it's now in the perfect form for us to find the critical points. We've taken a significant step towards solving the inequality, and we're well on our way to finding the solution set. So, let's keep the momentum going and move on to the next step!
Step 2: Finding the Critical Points
Alright, team, we've simplified the inequality to X² + 4X - 12 < 0. Now comes a crucial part: finding the critical points. These points are like the signposts on our mathematical journey, marking where the expression X² + 4X - 12 could change its sign (from positive to negative or vice versa). They're the values of X that make the expression equal to zero. Think of it like finding the exact spots where a rollercoaster goes from climbing uphill to plunging downhill – those spots are critical for understanding the ride's behavior. To find these critical points, we need to solve the equation X² + 4X - 12 = 0. This is a quadratic equation, and we have a couple of ways to tackle it. We can try factoring the quadratic expression, or if that seems tricky, we can use the quadratic formula. Factoring is often quicker if the expression factors nicely, but the quadratic formula always works, no matter what. Once we have our critical points, we'll use them to divide the number line into intervals, which will help us determine the solution to the inequality. So, let's roll up our sleeves and find those critical points!
Factoring the Quadratic Expression
Let's start by trying to factor the quadratic expression X² + 4X - 12. Factoring is like reverse-multiplying; we're trying to find two binomials that, when multiplied together, give us our quadratic expression. To do this, we need to find two numbers that multiply to give us the constant term (-12) and add up to give us the coefficient of the X term (4). Think of it as a puzzle where the pieces need to fit just right. Let's list the factors of -12: 1 and -12, -1 and 12, 2 and -6, -2 and 6, 3 and -4, and -3 and 4. Now, let's see which of these pairs adds up to 4. Looking at the list, we can see that -2 and 6 fit the bill perfectly: -2 * 6 = -12 and -2 + 6 = 4. So, we can factor our quadratic expression as (X - 2)(X + 6). This means that the equation X² + 4X - 12 = 0 can be rewritten as (X - 2)(X + 6) = 0. Now, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve: X - 2 = 0 and X + 6 = 0. These are simple linear equations, and we can solve them easily. So, let's move on and find the solutions for X!
Solving for X
We've factored our quadratic equation to (X - 2)(X + 6) = 0, and now it's time to solve for X. As we mentioned earlier, the zero-product property tells us that if the product of two factors is zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for X. First, let's take the factor (X - 2). Setting it equal to zero gives us the equation X - 2 = 0. To solve for X, we simply add 2 to both sides of the equation: X - 2 + 2 = 0 + 2, which simplifies to X = 2. So, one of our critical points is X = 2. Next, let's take the factor (X + 6). Setting it equal to zero gives us the equation X + 6 = 0. To solve for X, we subtract 6 from both sides of the equation: X + 6 - 6 = 0 - 6, which simplifies to X = -6. So, our other critical point is X = -6. We've found our critical points: X = 2 and X = -6. These are the values of X where the expression X² + 4X - 12 equals zero. Now, we'll use these critical points to divide the number line into intervals and test each interval to see if it satisfies our original inequality. So, let's keep going – we're getting closer to the solution!
Step 3: Testing the Intervals
Great job, guys! We've found our critical points, X = -6 and X = 2. These points are super important because they divide the number line into intervals, and these intervals are where our solution lies. Think of the number line as a road, and the critical points are the toll booths. We need to check each section of the road to see if it leads to our destination. So, how do we do that? We'll pick a test value from each interval and plug it into our simplified inequality X² + 4X - 12 < 0. If the test value makes the inequality true, then that entire interval is part of our solution. If it makes the inequality false, then that interval is not part of our solution. It's like a mathematical treasure hunt, and we're using test values as our clues. The critical points divide the number line into three intervals: (-∞, -6), (-6, 2), and (2, ∞). We'll pick a test value from each interval, plug it into the inequality, and see what happens. This might seem a bit tedious, but it's a systematic way to find the solution, and it's much easier than guessing! So, let's get testing and uncover the intervals that satisfy our inequality!
Choosing Test Values
Okay, let's choose our test values for each interval. Remember, we have three intervals: (-∞, -6), (-6, 2), and (2, ∞). We need to pick a number from each interval that's easy to work with. For the interval (-∞, -6), we can choose a number that's less than -6. A good choice would be X = -7. It's a nice, round number that's easy to plug into our inequality. For the interval (-6, 2), we need a number that's between -6 and 2. The easiest choice here is X = 0. Zero is always a great number to test because it simplifies calculations. Finally, for the interval (2, ∞), we need a number that's greater than 2. Let's go with X = 3. Again, it's a simple, round number that will make our calculations straightforward. So, we have our test values: X = -7 for the interval (-∞, -6), X = 0 for the interval (-6, 2), and X = 3 for the interval (2, ∞). Now, we're ready to plug these values into our simplified inequality X² + 4X - 12 < 0 and see which intervals satisfy the inequality. Remember, our goal is to find the intervals where the expression is less than zero. So, let's put on our detective hats and get testing!
Evaluating the Inequality
Now comes the fun part: plugging in our test values and seeing what happens! We'll start with X = -7 and substitute it into our inequality X² + 4X - 12 < 0. This gives us (-7)² + 4(-7) - 12 < 0. Let's simplify: 49 - 28 - 12 < 0, which simplifies further to 9 < 0. Hmm, 9 is definitely not less than 0, so this inequality is false for X = -7. This means the interval (-∞, -6) is not part of our solution. Next, let's try X = 0. Plugging it into our inequality gives us (0)² + 4(0) - 12 < 0. This simplifies to 0 + 0 - 12 < 0, which is -12 < 0. This is true! -12 is indeed less than 0. So, the interval (-6, 2) is part of our solution. Finally, let's test X = 3. Substituting it into our inequality gives us (3)² + 4(3) - 12 < 0. This simplifies to 9 + 12 - 12 < 0, which is 9 < 0. Again, this is false. 9 is not less than 0, so the interval (2, ∞) is not part of our solution. We've evaluated the inequality for all three test values, and we've found that only the interval (-6, 2) satisfies the inequality. This is a huge step forward! We're now ready to write the solution in interval notation, which will give us a clear and concise answer.
Step 4: Writing the Solution
Alright, guys, we've reached the final step! We've done all the hard work: simplifying the inequality, finding the critical points, and testing the intervals. Now, we just need to put it all together and write our solution in interval notation. This is like putting the final piece in a jigsaw puzzle – it completes the picture and gives us the full answer. From our testing, we found that the interval (-6, 2) is the only one that satisfies the inequality X² + 4X - 12 < 0. This means that all the values of X between -6 and 2 (but not including -6 and 2) make the inequality true. So, how do we write this in interval notation? Remember, interval notation uses parentheses and brackets to indicate whether the endpoints are included in the interval or not. Since our inequality is a strict inequality (<), we don't include the endpoints. This means we'll use parentheses. The interval notation for the solution is (-6, 2). This notation tells us that the solution includes all real numbers greater than -6 and less than 2. We've done it! We've successfully solved the polynomial inequality (X+5)(X-1) < 7, and we've expressed the solution clearly and concisely in interval notation. Give yourselves a big round of applause – you've earned it!
Interval Notation Explained
Let's take a moment to make sure we're all crystal clear on what interval notation means. It's a super handy way to represent a set of numbers, especially when we're dealing with inequalities. Think of it as a mathematical shorthand that allows us to communicate solutions quickly and accurately. In interval notation, we use parentheses ( ) and brackets [ ] to indicate the endpoints of an interval. Parentheses mean that the endpoint is not included in the interval, while brackets mean that the endpoint is included. For example, the interval (a, b) represents all real numbers between a and b, but not including a and b. The interval [a, b] represents all real numbers between a and b, including a and b. We also use infinity symbols ∞ and -∞ to represent intervals that extend without bound. Since infinity isn't a specific number, we always use parentheses with infinity symbols. So, the interval (a, ∞) represents all real numbers greater than a, and the interval (-∞, b) represents all real numbers less than b. In our solution (-6, 2), the parentheses tell us that -6 and 2 are not included in the solution. The solution includes all the numbers between -6 and 2, but not -6 and 2 themselves. Understanding interval notation is key to expressing solutions to inequalities clearly and correctly. It's a fundamental tool in mathematics, and mastering it will make your problem-solving skills even stronger. So, let's keep practicing and get comfortable with this essential notation!
Conclusion: Mastering Polynomial Inequalities
Woo-hoo! We've reached the end of our journey, and what a journey it has been! We've successfully navigated the world of polynomial inequalities, and we've conquered the challenge of solving (X+5)(X-1) < 7. You guys have been fantastic, sticking with me through each step, and now you have a solid understanding of how to tackle these types of problems. Let's take a quick recap of what we've covered. We started by simplifying the inequality, expanding the product, and rearranging the terms. Then, we found the critical points by factoring the quadratic expression and solving for X. After that, we tested the intervals created by the critical points to determine which intervals satisfy the inequality. Finally, we expressed our solution in interval notation: (-6, 2). But the real magic isn't just about getting the right answer to this specific problem. It's about the problem-solving skills you've developed along the way. You've learned how to break down a complex problem into smaller, manageable steps. You've learned the importance of accuracy and attention to detail. And you've learned how to use key mathematical concepts like the FOIL method, factoring, and interval notation. These skills will serve you well in all your mathematical endeavors and beyond. So, keep practicing, keep exploring, and keep pushing your boundaries. You've got this! Polynomial inequalities might have seemed daunting at first, but now you know that with the right approach and a bit of perseverance, you can master them. Congratulations on your achievement, and happy problem-solving!
Further Practice and Resources
Now that you've mastered the basics of solving polynomial inequalities, it's time to level up your skills! Practice makes perfect, so the more problems you tackle, the more confident you'll become. One great way to improve is to find additional practice problems online or in your textbook. Look for problems with varying levels of difficulty, so you can challenge yourself and continue to grow. Another fantastic resource is online math platforms like Khan Academy or Mathway. These platforms offer step-by-step solutions and explanations, which can be incredibly helpful if you get stuck. They also provide a wealth of practice problems and quizzes to test your understanding. Don't hesitate to revisit this guide or other resources whenever you need a refresher. Remember, learning is a continuous process, and it's okay to ask for help or review concepts as needed. You can also try creating your own polynomial inequalities and solving them. This is a great way to deepen your understanding and develop your problem-solving intuition. And don't forget to discuss these concepts with your classmates or friends. Explaining a concept to someone else is one of the best ways to solidify your own understanding. So, keep practicing, keep exploring, and keep learning. The world of mathematics is vast and fascinating, and there's always something new to discover. Happy learning, and remember, you've got the skills and the knowledge to conquer any mathematical challenge that comes your way!
