Solving Cubic Inequality: Step-by-Step Guide

Hey everyone! Today, we're diving into a cool math problem: solving the cubic inequality $2x^3 - 6x + 7 - (6x^2 - 4x + 1) > 0$. Don't worry, it looks a bit intimidating at first glance, but we'll break it down step-by-step to make it super easy to understand. We'll find the solution and express it in interval notation. So, grab your pencils and let's get started!

Understanding the Problem: The Cubic Inequality

Alright, so what exactly are we dealing with? Well, we have a cubic inequality. That means our highest power of x is 3. The goal here is to figure out for which values of x this inequality holds true. In other words, we're looking for the values of x that will make the left side of the inequality greater than zero. This might seem like a bit of a puzzle at first, but trust me, with the right approach, it's totally solvable.

To kick things off, let's simplify our inequality. First, we'll get rid of those parentheses by distributing the negative sign. This is a crucial step to make our equation more manageable and avoid any silly mistakes along the way. Remember, when you're distributing a negative, it changes the sign of each term inside the parentheses. So, after distributing, our inequality becomes: $2x^3 - 6x + 7 - 6x^2 + 4x - 1 > 0$. We're essentially rewriting the problem so we can work with it more effectively.

Now, let's rearrange the terms. We'll group the terms by their powers of x, starting with the highest power. This helps us to see the structure of the equation better and makes it easier to spot any patterns or simplifications we can make. This makes it easier to spot any terms we can combine. So, our inequality now looks like this: $2x^3 - 6x^2 - 2x + 6 > 0$. See how we've reorganized the terms to make things clearer? The goal here is to make the equation less cluttered and more organized.

After simplifying and rearranging, our inequality is in a much more user-friendly format. The key here is not to rush. Take your time, focus on each step, and double-check your work to avoid errors. This careful approach will set you up for success in solving the cubic inequality. By doing this, we make sure we have a solid, error-free foundation as we continue to solve the problem. Remember, in math, it's all about precision. Doing these simple steps with care goes a long way. This is a really important step.

Factoring and Finding the Roots

Now that we've simplified our cubic inequality, the next step is to tackle factoring. Factoring helps us to break down our equation into smaller, more manageable parts. These parts will give us insights into the behavior of the equation, especially where it crosses the x-axis (the roots). Finding the roots is a critical step because they act as boundary points that split the number line into intervals. In each interval, the inequality either holds true or false.

Let's get down to it. We need to factor the expression $2x^3 - 6x^2 - 2x + 6$. We can start by looking for common factors in the terms. Notice that all the terms have a factor of 2. So, we can factor out a 2: $2(x^3 - 3x^2 - x + 3) > 0$. Now we've simplified the equation a bit. We're on the right track!

Next, we'll look at factoring the cubic polynomial inside the parentheses. This might seem tricky, but we can use a method called factoring by grouping. This involves grouping the terms in pairs and then finding common factors in each pair. Here's how it works:

1. Group the first two terms and the last two terms: $(x^3 - 3x^2) + (-x + 3)$.
2. Factor out a common factor from each group. In the first group, we can factor out $x^2$, and in the second group, we can factor out -1: $x^2(x - 3) - 1(x - 3)$.
3. Now, we see that $(x - 3)$ is a common factor. Factor it out: $(x - 3)(x^2 - 1)$.

So, our factored expression becomes $2(x - 3)(x^2 - 1) > 0$. We're making good progress here!

But we are not done yet, because $(x^2 - 1)$ is also a difference of squares and can be factored further as $(x + 1)(x - 1)$. This is a common pattern that you'll see often in math problems. Applying it here will give us the fully factored form: $2(x - 3)(x + 1)(x - 1) > 0$. Now we have the inequality in a much simpler form. This helps us see the roots of the equation.

Factoring is so important because it reveals the roots of our equation, which are the values of x where the expression equals zero. For our factored expression, the roots are x = 3, x = -1, and x = 1. These are the critical points that we will use to find the solution intervals.

Determining the Intervals

We've found the roots, and now it's time to figure out which intervals satisfy the inequality. The roots we found, which are -1, 1, and 3, act as boundary points on the number line. These points divide the number line into intervals. The sign of the expression $2(x - 3)(x + 1)(x - 1)$ will either be positive or negative within each of these intervals. By testing values within each interval, we can determine the solution set.

Here’s a breakdown of the intervals we need to test:

1. Interval 1: $x < -1$
2. Interval 2: $-1 < x < 1$
3. Interval 3: $1 < x < 3$
4. Interval 4: $x > 3$

To test each interval, we'll choose a test value within each interval and substitute it into our factored inequality $2(x - 3)(x + 1)(x - 1) > 0$. If the result is positive, the inequality holds true for that interval. If the result is negative, the inequality does not hold true.

Let’s start testing:

• Interval 1: $x < -1$. Choose x = -2. Substituting into our inequality: $2(-2 - 3)(-2 + 1)(-2 - 1) = 2(-5)(-1)(-3) = -30$. The result is negative, so this interval is not part of our solution.
• Interval 2: $-1 < x < 1$. Choose x = 0. Substituting into our inequality: $2(0 - 3)(0 + 1)(0 - 1) = 2(-3)(1)(-1) = 6$. The result is positive, so this interval is part of our solution.
• Interval 3: $1 < x < 3$. Choose x = 2. Substituting into our inequality: $2(2 - 3)(2 + 1)(2 - 1) = 2(-1)(3)(1) = -6$. The result is negative, so this interval is not part of our solution.
• Interval 4: $x > 3$. Choose x = 4. Substituting into our inequality: $2(4 - 3)(4 + 1)(4 - 1) = 2(1)(5)(3) = 30$. The result is positive, so this interval is part of our solution.

We now have the results for each interval and can pinpoint the solution. The interval where the inequality is true is where the expression is greater than zero. The intervals that satisfy the inequality are $-1 < x < 1$ and $x > 3$. These are the solutions where our original inequality holds true. This is the final and crucial step.

Expressing the Solution in Interval Notation

Okay, guys! We're in the home stretch now. We've simplified, factored, and tested intervals. It's time to put it all together and present our solution in interval notation. Interval notation is just a way of expressing the set of all possible values of x that satisfy our inequality. This notation uses parentheses and brackets to show whether the endpoints are included in the solution.

Based on our interval testing, we found that the inequality holds true for two intervals: $-1 < x < 1$ and $x > 3$. Let's convert these into interval notation.

• For the interval $-1 < x < 1$, we use parentheses because the values -1 and 1 are not included in the solution (the inequality is strictly greater than, not greater than or equal to). In interval notation, this is written as $(-1, 1)$.
• For the interval $x > 3$, we also use a parenthesis at 3 because 3 is not included. Since the interval goes to infinity, we use infinity, which is always written with a parenthesis. In interval notation, this is written as $(3, ext{∞})$.

To express the complete solution, we combine these two intervals using the union symbol (∪). The union symbol tells us that the solution includes either the first interval OR the second interval. So, the final answer in interval notation is $(-1, 1) ∪ (3, ext{∞})$. This notation tells us that the inequality holds true for all x values between -1 and 1 (excluding -1 and 1), and for all x values greater than 3 (excluding 3). Congratulations, we solved the inequality!

This format is a concise and standard way to represent the solution set. Understanding interval notation is super important in math, because it simplifies things, and it is a universally recognized way of communicating solutions. When expressing your answers in math, using the standard format is always a good idea, as it makes your work clear and easy to understand.

Conclusion: Wrapping Things Up

Alright, folks, that's a wrap! We've successfully solved the cubic inequality $2x^3 - 6x + 7 - (6x^2 - 4x + 1) > 0$. We used simplification, factoring, interval testing, and finally, we expressed our solution in interval notation. The final solution is $(-1, 1) ∪ (3, ext{∞})$.

Remember, the key to solving inequalities like these is to break them down into manageable steps and to double-check your work along the way. Factoring, understanding the roots, and interval testing are essential skills. And don't forget the importance of using the right notation to present your solutions clearly.

Keep practicing, keep learning, and don't be afraid to tackle new challenges. You've got this! If you liked this guide, give it a thumbs up. And feel free to share any math problems you'd like me to solve in the comments below. Thanks for joining me, and I'll see you in the next one!