Solving The Inequality: A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of inequalities and tackle the problem: $\frac{x-2}{x^2(x+1)}>0$. Don't worry, it might look a little intimidating at first, but we'll break it down step-by-step to make it super clear. This type of problem is super common in math and understanding how to solve it is a valuable skill. We'll explore the intervals where this inequality holds true, figuring out those sneaky values that make the expression positive. Think of it like a treasure hunt, where we're looking for the 'x' values that unlock the positive side of this mathematical puzzle. Ready to crack the code? Let's get started!

Understanding the Problem: The Inequality Demystified

Okay, so the core of our problem is the inequality: $\frac{x-2}{x^2(x+1)}>0$. This means we're trying to find all the values of x that make the expression on the left-hand side greater than zero (positive). Remember, a fraction is positive if either both the numerator and denominator are positive, or both are negative. The real trick here is to identify the critical points where the expression can change signs, like when the numerator or denominator equals zero. These points will divide the number line into intervals that we'll need to investigate. In this case, we have a rational expression, so we need to be extra careful about the denominator. Remember, division by zero is a big no-no! So, before we even start, let's list the values of x that make the denominator zero. These are the values we must exclude from our solution. So, our main goal is to solve the inequality and determine the intervals where the function is positive. By understanding the core concept, you'll be well-equipped to tackle more complex inequalities in the future.

Let's break down the components. The numerator is $(x-2)$, which becomes zero when $x = 2$. The denominator is $x^2(x+1)$. The factor $x^2$ becomes zero when $x=0$, and the factor $(x+1)$ becomes zero when $x = -1$. Therefore, our critical points are $x = -1, 0,$ and $2$. These are the points that will divide our number line into intervals. Always remember to consider these points; they are the keys to unlocking the solution. Make sure to clearly mark these critical points on the number line to visualize the intervals effectively. It's like setting up the chessboard before the game begins; we're preparing everything for our analysis. Now that we have our critical points, let's explore how to find the intervals of our solution.

Identifying Critical Points and Their Significance

As we previously discussed, the critical points are the values of x that make either the numerator or the denominator equal to zero. These are the points where the expression can potentially change its sign (from positive to negative, or vice versa). For our inequality, the critical points are $x = -1, x = 0,$ and $x = 2$. Keep in mind that these points aren't necessarily solutions to the inequality itself. The roots of the numerator and the denominator's factors are important because they are where the sign of the expression can change. However, because our inequality is strict ('>' instead of '>='), we do not include the roots of the numerator in our solution. This means that at $x = 2$, the expression is zero, which does not satisfy the inequality $\frac{x-2}{x^2(x+1)}>0$. Also, we must exclude any values that make the denominator zero, as division by zero is undefined. This means that $x$ cannot equal $-1$ or $0$. These critical points, therefore, create the boundaries for our intervals.

Here’s a quick recap of why these points matter so much: at the critical points, the numerator or denominator changes sign or becomes undefined. This is where we will check the value of x using the number line and the sign analysis methods. By understanding the behavior of the expression around these critical points, we can determine the intervals where the inequality holds true. These points help in solving the inequality and determining the correct solution set. Always be extra careful with them! So, let's get those critical points on the number line.

Analyzing the Intervals: Testing the Waters

Now that we've identified our critical points, it's time to test the intervals created by these points. These are: $(-\,\infty, -1)$, $(-1, 0)$, $(0, 2)$, and $(2, \,\infty)$. We'll pick a test value within each interval and substitute it into our original inequality, $\frac{x-2}{x^2(x+1)}>0$, to see if it holds true. This is the heart of the solution process: finding the ranges where the function is positive. The sign analysis method involves selecting test values within each interval and evaluating the sign of the expression at those points. This methodical approach will let us pinpoint the solution intervals with confidence. Remember, the sign of the overall expression is what matters. Let's get started, shall we?

Interval 1: $(-\infty, -1)$

Let's choose a test value, say $x = -2$. Substitute it into the inequality:

$\frac{(-2)-2}{(-2)^2((-2)+1)} = \frac{-4}{4(-1)} = \frac{-4}{-4} = 1 > 0$

Since the result is positive, the inequality holds true for this interval. Therefore, $(-\infty, -1)$ is part of our solution.

Interval 2: $(-1, 0)$

Let's choose a test value, say $x = -0.5$. Substitute it into the inequality:

$\frac{(-0.5)-2}{(-0.5)^2((-0.5)+1)} = \frac{-2.5}{0.25(0.5)} = \frac{-2.5}{0.125} = -20 < 0$

Since the result is negative, the inequality does not hold true for this interval. Therefore, $(-1, 0)$ is not part of our solution.

Interval 3: $(0, 2)$

Let's choose a test value, say $x = 1$. Substitute it into the inequality:

$\frac{(1)-2}{(1)^2((1)+1)} = \frac{-1}{1(2)} = \frac{-1}{2} = -0.5 < 0$

Since the result is negative, the inequality does not hold true for this interval. Therefore, $(0, 2)$ is not part of our solution.

Interval 4: $(2, \infty)$

Let's choose a test value, say $x = 3$. Substitute it into the inequality:

$\frac{(3)-2}{(3)^2((3)+1)} = \frac{1}{9(4)} = \frac{1}{36} > 0$

Since the result is positive, the inequality holds true for this interval. Therefore, $(2, \infty)$ is part of our solution. We've now checked all the intervals. Nice one, guys!

Solution Set: Putting It All Together

Alright, we've tested all our intervals, and now we know where the inequality is true! Based on our analysis, the solution to the inequality $\frac{x-2}{x^2(x+1)}>0$ is $(-\infty, -1) \cup (2, \infty)$. Remember, we exclude the points where the denominator is zero (x = -1 and x = 0) and the root of the numerator (x = 2). The solution set represents all the values of x for which the original inequality is satisfied. Here's how we've arrived at the solution. By breaking down the problem, carefully identifying critical points, and systematically testing intervals, we've found the range of x values that meet the criteria. Always make sure to consider each step, as each plays a vital role. Writing the solution in interval notation ensures we accurately represent all the x values that make our inequality true.

So, to recap, the inequality $\frac{x-2}{x^2(x+1)}>0$ holds true for $x$ in the interval $(-\infty, -1)$ and $(2, \infty)$. Keep practicing and you will be a master of solving inequalities in no time. Congratulations! You've successfully navigated through the problem and arrived at the solution. This is a powerful technique that you can apply to various mathematical problems.

Conclusion: Mastering Inequalities

Great job, everyone! You've successfully solved the inequality $\frac{x-2}{x^2(x+1)}>0$. We started by understanding the problem, finding critical points, testing intervals, and finally, assembling our solution set. This method is a solid framework that you can apply to various inequality problems, not just rational ones. Remember, understanding inequalities is a fundamental skill in math. With a bit of practice and patience, you can confidently tackle these problems and gain a deeper understanding of mathematical concepts. Remember the key takeaways: identify critical points, test intervals, and express your solution accurately. Keep practicing, and you'll become a pro at solving inequalities. Keep up the amazing work! Don't hesitate to revisit these steps anytime you face a similar problem. And hey, you've got this! Math can be fun and rewarding, and with the right approach, you can conquer any challenge. Keep practicing and applying these techniques, and you'll be well on your way to mathematical mastery.