Solving $x^2 + 5x < 14$: Interval Notation Explained

Hey guys! Today, we're diving into a classic algebra problem: solving the inequality $x^2 + 5x < 14$ and expressing the solution in interval notation. This is a fundamental concept in mathematics, and understanding it is crucial for more advanced topics. So, let's break it down step by step and make sure we all get it.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We have a quadratic inequality, $x^2 + 5x < 14$. Our goal is to find all the values of x that make this inequality true. Now, remember that simply solving for x as if it were an equation won't work here because we're dealing with an inequality. The solution will be a range of values, not just one or two specific numbers. And, we need to express that range using interval notation, which is a specific way of writing sets of numbers using parentheses and brackets. It might seem daunting at first, but don't worry, we'll get there together!

Think of it like finding the sweet spot – the values of x that fit perfectly into our inequality. Inequalities are like picky eaters; they only accept certain values! The key here is to first manipulate the inequality into a standard form, then find the critical points where the expression equals zero, and finally test intervals to determine where the inequality holds true. We're essentially mapping out the number line and identifying the sections that satisfy our condition. This involves a bit of algebraic maneuvering and careful consideration of the inequality sign, but once you get the hang of it, it becomes almost second nature. So, let’s roll up our sleeves and get started on solving this quadratic inequality.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this inequality step-by-step. It's like following a recipe – each step is important to get the final delicious result!

1. Rearrange the Inequality

The first thing we need to do is rearrange the inequality so that we have zero on one side. This is a crucial step because it sets us up to find the roots of the quadratic expression. So, we subtract 14 from both sides of the inequality:

$x^2 + 5x - 14 < 0$

Now we have a quadratic expression on the left side and zero on the right. This is the standard form we need for solving quadratic inequalities. Think of it as leveling the playing field – we're making sure everything is in the right place before we proceed. This rearrangement allows us to easily identify the coefficients and constants, which will be helpful in the next steps, particularly when we factor the quadratic expression. So, make sure you get this step right – it's the foundation for everything else we're going to do!

2. Factor the Quadratic

Next up, we need to factor the quadratic expression $x^2 + 5x - 14$. Factoring is like cracking a code – we're trying to find two expressions that multiply together to give us our original quadratic.

We're looking for two numbers that multiply to -14 and add up to 5. Those numbers are 7 and -2. So, we can factor the quadratic as follows:

$(x + 7)(x - 2) < 0$

This is a super important step! Factoring transforms the inequality into a product of two terms, which makes it much easier to analyze. Each factor represents a line, and the sign of the product depends on the signs of the individual factors. This is where the magic happens – we're turning a complex expression into something we can easily understand and work with. If you're not comfortable with factoring, it's worth brushing up on those skills because it's a fundamental tool in algebra and calculus.

3. Find the Critical Points

Now we need to find the critical points. These are the values of x that make the expression $(x + 7)(x - 2)$ equal to zero. In other words, they're the roots of the equation $(x + 7)(x - 2) = 0$.

To find them, we set each factor equal to zero and solve for x:

$x + 7 = 0 => x = -7$

$x - 2 = 0 => x = 2$

So our critical points are x = -7 and x = 2. These points are like the boundaries of our solution. They divide the number line into intervals where the expression $(x + 7)(x - 2)$ will be either positive or negative. Think of them as checkpoints – they're the places where the expression might change its behavior. These critical points are super important because they help us determine the intervals where the inequality is satisfied. Without them, we'd be wandering in the dark, trying to guess the solution. So, make sure you identify these critical points accurately – they're the key to unlocking the solution!

4. Test Intervals

The critical points divide the number line into three intervals: $(-\infty, -7)$, $(-7, 2)$, and $(2, \infty)$. We need to test a value from each interval in the inequality $(x + 7)(x - 2) < 0$ to see if it holds true.

• Interval $(-\infty, -7)$: Let's test x = -8:

  $(-8 + 7)(-8 - 2) = (-1)(-10) = 10 > 0$ (False)

• Interval $(-7, 2)$: Let's test x = 0:

  $(0 + 7)(0 - 2) = (7)(-2) = -14 < 0$ (True)

• Interval $(2, \infty)$: Let's test x = 3:

  $(3 + 7)(3 - 2) = (10)(1) = 10 > 0$ (False)

This is the heart of solving the inequality! We're essentially playing detective, testing different suspects (intervals) to see who's the culprit (satisfies the inequality). By plugging in test values, we can determine whether the expression is positive or negative in each interval. This is crucial because we're looking for the interval(s) where the expression is less than zero. Think of it like a trial – each interval gets its chance to prove its innocence (or guilt). And the outcome of these tests will tell us the solution to the inequality. So, make sure you choose your test values wisely and perform the calculations carefully – the fate of the inequality rests in your hands!

5. Write the Solution in Interval Notation

The inequality $(x + 7)(x - 2) < 0$ is true for the interval $(-7, 2)$. Notice that we use parentheses instead of brackets because the inequality is strictly less than zero, meaning that the endpoints -7 and 2 are not included in the solution.

So, the solution in interval notation is $(-7, 2)$.

We did it! We've successfully navigated the twists and turns of this quadratic inequality and arrived at our destination: the interval notation solution. This final step is like putting the cherry on top of a sundae – it's the perfect way to present our answer. Interval notation is a concise and elegant way of expressing a range of values, and it's essential for communicating mathematical solutions clearly. Remember, parentheses indicate that the endpoints are not included, while brackets would mean they are. In this case, we use parentheses because the inequality is strictly less than, not less than or equal to. So, pat yourself on the back – you've mastered another crucial concept in algebra!

Why Interval Notation Matters

You might be wondering, "Why all this fuss about interval notation?" Well, it's not just some fancy way of writing things down. Interval notation is a powerful tool that helps us clearly and concisely express solutions to inequalities and domains of functions. It's a universal language in mathematics, allowing mathematicians and students alike to communicate complex ideas effectively. Think of it as a shorthand – instead of writing out long, wordy descriptions of intervals, we can simply use a compact notation that everyone understands.

Moreover, interval notation is particularly useful in calculus and other advanced topics. When dealing with limits, continuity, and derivatives, we often need to work with intervals. Interval notation makes these operations much smoother and less prone to errors. It's like having a well-organized toolbox – when you need to grab a specific tool, you know exactly where to find it. Similarly, when you need to express a solution set, interval notation is the perfect tool for the job. So, mastering this notation is not just about solving problems; it's about building a solid foundation for your future mathematical endeavors. It's a skill that will pay off dividends as you delve deeper into the world of mathematics.

Common Mistakes to Avoid

Solving inequalities can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

• Forgetting to rearrange the inequality: Always make sure you have zero on one side before factoring.
• Incorrectly factoring the quadratic: Double-check your factors to make sure they multiply back to the original expression.
• Using brackets instead of parentheses (or vice versa): Remember that parentheses indicate that the endpoints are not included, while brackets mean they are included. This is a subtle but crucial distinction. A little slip here can change the entire meaning of your solution.
• Not testing intervals: Don't just assume the solution is the interval between the critical points. You must test values in each interval to be sure.
• Dividing or multiplying by a negative number without flipping the inequality sign: This is a classic mistake! Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Think of these mistakes as traps – they're easy to fall into if you're not paying attention. But with a little awareness and careful checking, you can avoid them and ensure you get the correct answer. It's like being a detective – you need to look for clues and potential pitfalls to solve the case. So, keep these common mistakes in mind, and you'll be well on your way to becoming an inequality-solving pro!

Practice Makes Perfect

The best way to master solving inequalities and using interval notation is to practice! Try working through different examples with varying degrees of difficulty. The more you practice, the more comfortable you'll become with the process. Think of it like learning a musical instrument – you wouldn't expect to become a virtuoso overnight. It takes time, dedication, and consistent practice. Similarly, with math, the more you engage with the material, the better you'll understand it.

Look for practice problems in your textbook, online, or ask your teacher for additional exercises. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from your mistakes and keep pushing forward. And remember, there are tons of resources available to help you – from online tutorials to study groups to your friendly neighborhood math teacher. So, don't hesitate to reach out for support when you need it. With consistent effort and practice, you'll be solving inequalities like a pro in no time!

Conclusion

So, to answer the question, the solution to the inequality $x^2 + 5x < 14$ in interval notation is $(-7, 2)$. We've walked through the steps together, from rearranging the inequality to testing intervals and finally expressing the solution in interval notation. Hopefully, this has demystified the process and given you a solid understanding of how to tackle these types of problems.

Remember, math is like building a house – each concept builds upon the previous one. Mastering the fundamentals, like solving inequalities and using interval notation, will set you up for success in more advanced topics. So, keep practicing, keep asking questions, and most importantly, keep believing in yourself. You've got this! And hey, if you ever get stuck, just remember this guide, and you'll be well on your way to solving any inequality that comes your way. Happy solving, guys!