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

Hey guys, let's dive into solving inequalities and expressing the solutions using interval notation. Today, we're tackling the inequality $x^2+5x<14$. This might look a little intimidating at first, but trust me, once we break it down step-by-step, it'll be as clear as day. We'll explore how to find the values of $x$ that make this statement true and then represent those solutions in a super handy format called interval notation. So, grab your notebooks, maybe a cup of coffee, and let's get this math party started!

Understanding the Inequality $x^2+5x<14$

Alright, so the core of our problem is the inequality $x^2+5x<14$. This isn't just some random jumble of numbers and symbols; it's asking us to find all the possible values for '$x$' that make the expression on the left side ($x^2+5x$) strictly less than the number on the right side (14). Think of it like a balancing act. We want to find where the $x^2+5x$ side is lighter than the 14 side. To do this effectively, the first crucial step is to get all the terms on one side of the inequality. This means we need to subtract 14 from both sides, giving us a zero on the right. So, our inequality transforms into $x^2+5x-14<0$. This form is super important because it allows us to treat this as a related equation, $x^2+5x-14=0$, to find the boundary points where the expression might change its sign (from positive to negative, or vice versa). These boundary points are critical for determining the intervals where our original inequality holds true. We're essentially looking for the 'x-intercepts' if we were to graph the related quadratic function $y = x^2+5x-14$. Understanding this transformation is key to mastering how to solve quadratic inequalities. It's all about setting up the problem correctly so that we can find those crucial turning points. Remember, the goal is to isolate the inequality and set it equal to zero to find the roots, which will then define our test intervals.

Finding the Critical Points

Now that we have our inequality in the form $x^2+5x-14<0$, the next logical step is to find the critical points. These are the values of $x$ where the expression $x^2+5x-14$ equals zero. Why are these important? Because they are the points where the expression might change from being positive to negative, or negative to positive. For our inequality $x^2+5x-14<0$, we are interested in where the expression is negative. The critical points act as dividers, splitting the number line into distinct intervals. We can find these critical points by solving the related quadratic equation: $x^2+5x-14=0$. Now, how do we solve this? We can try factoring, using the quadratic formula, or completing the square. Let's try factoring first, as it's often the quickest method if it works. We're looking for two numbers that multiply to -14 and add up to +5. Let's think about the factors of 14: (1, 14) and (2, 7). To get a product of -14, one factor must be positive and the other negative. To get a sum of +5, the larger factor should be positive. So, if we try -2 and +7, their product is $(-2)*(7) = -14$, and their sum is $-2+7 = 5$. Perfect! So, we can factor the quadratic as $(x-2)(x+7)=0$. Now, for this product to be zero, at least one of the factors must be zero. This gives us two possibilities: $x-2=0$ or $x+7=0$. Solving these simple linear equations, we get $x=2$ and $x=-7$. These are our critical points! They divide the number line into three potential intervals: $(- extit{infinity}, -7)$, $(-7, 2)$, and $(2, extit{infinity})$. We'll use these points to test which intervals satisfy our original inequality $x^2+5x-14<0$. It's really about identifying those key values that could potentially change the truthiness of our inequality.

Testing the Intervals

With our critical points $x=-7$ and $x=2$ identified, we've successfully divided the number line into three distinct intervals: $(- extit{infinity}, -7)$, $(-7, 2)$, and $(2, extit{infinity})$. Our next crucial step is to test a value from each of these intervals to see if it satisfies the original inequality, $x^2+5x-14<0$. Remember, the critical points themselves are not included because the inequality is strictly less than ($<$) zero, not less than or equal to ($\le$). Let's pick a test value for each interval:

1. Interval 1: $(- extit{infinity}, -7)$. Let's choose a nice, easy number that's less than -7, like $x=-10$. Now, we substitute this into our inequality: $(-10)^2 + 5(-10) - 14$. This gives us $100 - 50 - 14 = 50 - 14 = 36$. Is 36 less than 0? No, it's not. So, this interval is not part of our solution.

2. Interval 2: $(-7, 2)$. Let's pick a value between -7 and 2. How about $x=0$? It's super easy to plug in! So, we have $(0)^2 + 5(0) - 14$. This simplifies to $0 + 0 - 14 = -14$. Is -14 less than 0? Yes, it is! This interval is part of our solution.

3. Interval 3: $(2, extit{infinity})$. Let's pick a number greater than 2, say $x=5$. Plugging this in: $(5)^2 + 5(5) - 14$. This equals $25 + 25 - 14 = 50 - 14 = 36$. Is 36 less than 0? No, it's not. So, this interval is not part of our solution.

By testing these intervals, we've discovered that only the interval between -7 and 2 makes the inequality $x^2+5x-14<0$ true. This systematic testing is what allows us to pinpoint the exact range of values for $x$ that satisfy the original condition. It’s like being a detective, checking all the clues to find the one true path!

Expressing the Solution in Interval Notation

We've done the hard work, guys! We found that the only interval where $x^2+5x-14<0$ is true is between our critical points -7 and 2. Now comes the final, and arguably the coolest, part: expressing this solution using interval notation. This is a standard mathematical way to represent a range of numbers. Since our inequality is strictly less than zero ($<0$), we know that the endpoints, -7 and 2, are not included in our solution set. For intervals that do not include their endpoints, we use parentheses () instead of square brackets []. So, the interval between -7 and 2, excluding -7 and 2 themselves, is written as $(-7, 2)$. This notation tells us that $x$ can be any number greater than -7 and less than 2. It's a concise and precise way to communicate our findings. Looking back at the options provided:

• A. $[-7,2]$: This includes the endpoints, which is incorrect for a strict inequality.
• B. $(- extit{infinity},-7) extit{cup} (2, extit{infinity})$: This represents the intervals where the expression is greater than zero, which is the opposite of what we need.
• C. $(-7,2)$: This correctly represents the interval where the expression is less than zero, excluding the endpoints.
• D. $(- extit{infinity},-7] extit{cup} [2, extit{infinity})$: This includes the endpoints and represents where the expression is greater than or equal to zero.

Therefore, the correct way to express the solution of $x^2+5x<14$ in interval notation is $(-7, 2)$. This final step ensures that our mathematical answer is presented in the universally understood format of interval notation, making it easy for anyone to interpret. It's the perfect wrap-up to our inequality-solving adventure!

Conclusion: Mastering Quadratic Inequalities

So there you have it, fam! We've successfully tackled the quadratic inequality $x^2+5x<14$ from start to finish. We began by rearranging it into the standard form $x^2+5x-14<0$, found the critical points by solving the related equation $x^2+5x-14=0$ (which gave us $x=-7$ and $x=2$), and then systematically tested the intervals created by these points. Our testing revealed that the inequality holds true only for values of $x$ strictly between -7 and 2. Finally, we translated this range into the clear and concise interval notation $(-7, 2)$. This process of solving quadratic inequalities is a fundamental skill in algebra and opens the door to understanding more complex mathematical concepts. Remember, the key steps are: 1. Standardize the inequality (get zero on one side). 2. Find the critical points (solve the related equation). 3. Test intervals. 4. Write the solution in the required notation. Keep practicing these steps, and you'll become a whiz at these problems. Don't be afraid to pause, re-read, and work through examples. The more you practice, the more natural it becomes. Keep up the great work, and happy problem-solving!