Solving (x - 3) . (- X² + 3x + 10) > 0 With Inequalities A Comprehensive Guide

Hey there, math enthusiasts! Ever feel like inequalities are these cryptic puzzles that just need the right key to unlock? Well, you're in the right place! Today, we're diving deep into the world of 1st and 2nd-degree inequalities, armed with the knowledge to conquer even the trickiest ones. Our mission? To crack the solution set for the inequality (x - 3) . (- x² + 3x + 10) > 0 within the realm of real numbers. Buckle up, because this is going to be an exciting ride!

Delving into the Fundamentals of Inequalities

Before we tackle the main problem, let's take a moment to solidify our understanding of what inequalities are all about. Think of equations as these precise balances, where both sides are perfectly equal. Inequalities, on the other hand, are a bit more flexible. They describe relationships where one side is either greater than, less than, greater than or equal to, or less than or equal to the other side. These relationships are the core of inequalities, and understanding them is key to solving them effectively. To kick things off, remember those trusty number lines we used back in the day? They're about to become our best friends again! Number lines are fantastic tools for visualizing inequalities. Imagine plotting points on the line to represent solutions. When we're dealing with inequalities, we're not just looking for single points; we're often hunting for entire intervals or ranges of values that satisfy the given condition. This is where things get interesting! Solving inequalities involves a similar process to solving equations, but with a few crucial twists. We can add, subtract, multiply, and divide both sides, but there's one rule we absolutely cannot forget: When we multiply or divide both sides by a negative number, we must flip the inequality sign. This is super important because failing to do so will lead us down the wrong path. Now, let's break down the different types of inequalities we might encounter. First-degree inequalities, also known as linear inequalities, are those where the highest power of the variable is 1. Think of them as the straightforward, no-nonsense inequalities. Second-degree inequalities, or quadratic inequalities, involve variables raised to the power of 2. These are a bit more complex but equally manageable once we know the tricks. The real magic happens when we combine these concepts. Often, we'll encounter inequalities that involve products or quotients of expressions. To solve these, we need to analyze the signs of each factor and how they interact. This is where sign analysis comes into play, and it's a powerful technique we'll use extensively in our main problem. So, with these fundamentals in mind, we're ready to dive into the specifics of our problem and start unraveling its solution.

Unpacking the Inequality (x - 3) . (- x² + 3x + 10) > 0

Alright guys, let's zero in on the inequality at hand: (x - 3) . (- x² + 3x + 10) > 0. At first glance, it might seem a bit intimidating, but don't worry, we're going to break it down step by step. The core of solving this inequality lies in understanding how the product of two factors can be greater than zero. Remember, a product is positive if both factors are positive or if both factors are negative. This simple rule is our guiding light. Our inequality consists of two main factors: (x - 3) and (- x² + 3x + 10). To solve the inequality, we need to figure out the intervals where each factor is positive or negative. This is where our sign analysis skills come into play. Let's start with the first factor, (x - 3). This is a linear expression, and it's equal to zero when x = 3. For values of x less than 3, the factor (x - 3) will be negative, and for values of x greater than 3, it will be positive. Easy peasy, right? Now, let's tackle the second factor, (- x² + 3x + 10). This is a quadratic expression, which means we need to find its roots to determine where it changes sign. To find the roots, we set the expression equal to zero: - x² + 3x + 10 = 0. This is a quadratic equation that we can solve using the quadratic formula, factoring, or any other method you prefer. Let's use the quadratic formula for this one: x = [-b ± √(b² - 4ac)] / (2a). In our case, a = -1, b = 3, and c = 10. Plugging these values into the formula, we get: x = [-3 ± √(3² - 4(-1)(10))] / (2(-1)). Simplifying further, we have: x = [-3 ± √(9 + 40)] / (-2) x = [-3 ± √49] / (-2) x = [-3 ± 7] / (-2). This gives us two roots: x1 = (-3 + 7) / (-2) = -2 and x2 = (-3 - 7) / (-2) = 5. So, the quadratic expression (- x² + 3x + 10) equals zero when x = -2 and x = 5. These are our critical points. Now, we need to determine the sign of the quadratic expression in the intervals defined by these roots. Since the coefficient of the x² term is negative, the parabola opens downwards. This means that the expression is positive between the roots and negative outside the roots. So, (- x² + 3x + 10) is positive when -2 < x < 5 and negative when x < -2 or x > 5. With the signs of both factors determined, we're ready to create a sign chart to visualize the solution.

Crafting the Sign Chart for Solution Visualization

The sign chart is our secret weapon for conquering inequalities. It's a visual representation that helps us track the signs of each factor across different intervals, making it super clear where the overall inequality holds true. Think of it as a map guiding us to the solution treasure! To build our sign chart, we'll start by drawing a number line. On this line, we'll mark all the critical points we identified earlier. These are the points where either of our factors equals zero: x = -2, x = 3, and x = 5. These points divide the number line into intervals, and within each interval, the sign of each factor remains constant. Now, we'll create a table above the number line. The rows of the table will represent our factors: (x - 3) and (- x² + 3x + 10). The columns will represent the intervals created by our critical points: x < -2, -2 < x < 3, 3 < x < 5, and x > 5. For each factor and each interval, we'll determine the sign of the factor. We already did this in the previous section, so now we're just organizing our findings. For the factor (x - 3), we know it's negative when x < 3 and positive when x > 3. So, we'll fill in the sign chart accordingly. For the factor (- x² + 3x + 10), we know it's positive when -2 < x < 5 and negative when x < -2 or x > 5. Again, we'll fill in the sign chart based on these observations. Once we have the signs of each factor in each interval, we can determine the sign of the product (x - 3) . (- x² + 3x + 10). Remember, a product is positive if both factors have the same sign (both positive or both negative), and it's negative if the factors have opposite signs. So, we'll multiply the signs in each interval to find the sign of the product. Now comes the exciting part! We're looking for the intervals where the product (x - 3) . (- x² + 3x + 10) is greater than zero, meaning it's positive. We'll identify those intervals on our sign chart. These intervals represent the solution set for our inequality. But wait, there's a little detail we need to consider. Our inequality is strictly greater than zero, which means we don't include the points where the expression equals zero. These are our critical points: x = -2, x = 3, and x = 5. So, when we write our solution set, we'll use open intervals (parentheses) to indicate that these points are not included. With our sign chart complete and our solution intervals identified, we're ready to express the final answer in interval notation.

Expressing the Grand Finale: The Solution Set

Drumroll, please! We've reached the final stage of our inequality-solving adventure: expressing the solution set. After all the hard work we've put in analyzing factors, creating sign charts, and navigating intervals, it's time to present our findings in a clear and concise way. Interval notation is the language we'll use to communicate our solution set. It's a neat way of representing intervals of numbers using parentheses and brackets. Parentheses indicate that the endpoint is not included in the interval, while brackets indicate that it is. Remember, our inequality is (x - 3) . (- x² + 3x + 10) > 0, which means we're looking for the intervals where the expression is strictly greater than zero. This tells us that we'll be using parentheses for all our endpoints since we don't want to include the points where the expression equals zero. Looking back at our sign chart, we identified the intervals where the product is positive. These are the intervals that make up our solution set. Let's say we found that the product is positive in the intervals (-2, 3) and (5, ∞). This means that all values of x between -2 and 3, as well as all values of x greater than 5, satisfy our inequality. To express this in interval notation, we simply write: (-2, 3) ∪ (5, ∞). The symbol