Solving $x^4-3 X^3 < 6(x-3)$ Algebraically A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of inequalities, specifically how to solve them using algebraic methods. We'll be tackling a tricky one: $x^4 - 3x^3 < 6(x - 3)$. This isn't your everyday linear inequality, so we need to roll up our sleeves and get ready for some algebraic maneuvering. Solving inequalities algebraically can be a bit like navigating a maze, but with the right tools and strategies, we can find our way to the solution set. This guide will break down each step, ensuring you understand not just the how, but also the why behind the algebraic manipulations. We’ll explore how to rearrange the inequality, factor polynomials, and analyze intervals to determine where the inequality holds true. Understanding these techniques is crucial not only for solving this specific problem but also for tackling a wide range of mathematical challenges. So, let's jump in and unravel this inequality together!

Our mission, should we choose to accept it (and we do!), is to determine the solution set for the inequality:

$x^4 - 3x^3 < 6(x - 3)$.

Remember, the solution set represents all the values of x that make this inequality true. Unlike equations where we typically find specific values, inequalities often lead to intervals or ranges of values. This is where the algebraic approach becomes particularly powerful, allowing us to systematically explore the entire number line and identify the regions where the inequality is satisfied. The journey might seem daunting at first, with the fourth-degree polynomial and the presence of parentheses, but don't worry! We'll break it down step-by-step, using algebraic techniques to transform the inequality into a more manageable form. Think of it as a puzzle; each algebraic manipulation is a piece that brings us closer to the complete solution. We will focus on using algebraic manipulations to simplify, factor, and ultimately solve this inequality, and we'll show every step of the process so you can follow along.

The first thing we need to do when solving inequalities algebraically is to bring all the terms to one side, leaving zero on the other. This is similar to solving equations, where we aim to isolate the variable. By setting one side to zero, we can then analyze the sign of the expression on the other side. For our inequality, $x^4 - 3x^3 < 6(x - 3)$, we begin by distributing the 6 on the right side:

$x^4 - 3x^3 < 6x - 18$

Next, we subtract $6x$ and add $18$ to both sides to get everything on the left:

$x^4 - 3x^3 - 6x + 18 < 0$

Now we have a polynomial expression on the left side that we need to analyze. This is a crucial step because it transforms the inequality into a form that's easier to work with. By rearranging the terms, we've created a situation where we can focus on finding the values of x that make the polynomial expression less than zero. This sets the stage for the next step, which involves factoring the polynomial. Remember, our goal is to find the intervals where the expression is negative, and having it in factored form will make that process much simpler. This initial rearrangement is a fundamental technique in solving inequalities, providing a clear path forward to finding the solution set.

Factoring is a key technique in solving polynomial inequalities. It allows us to break down a complex expression into simpler components, making it easier to analyze its sign. Our expression is $x^4 - 3x^3 - 6x + 18$. Notice that we have four terms, which suggests that we might be able to use factoring by grouping. Let's group the first two terms and the last two terms:

$(x^4 - 3x^3) + (-6x + 18) < 0$

Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is $x^3$, and from the second group, it's $-6$:

$x^3(x - 3) - 6(x - 3) < 0$

We see a common factor of $(x - 3)$ in both terms, so we factor it out:

$(x - 3)(x^3 - 6) < 0$

Now we have factored the polynomial into a product of two terms. This is a significant step because it allows us to analyze the sign of the expression by considering the signs of each factor individually. Factoring is like dissecting a complex problem into smaller, manageable parts. By breaking down the polynomial, we can now identify the critical points where the expression might change its sign. These critical points, also known as zeros or roots, are the values of x that make either factor equal to zero. Finding these zeros is the next crucial step in determining the solution set for our inequality. The factored form provides a clear pathway to identifying these zeros and understanding how they divide the number line into intervals where the expression maintains a consistent sign.

The critical points are the values of x where the expression $(x - 3)(x^3 - 6)$ equals zero. These points are crucial because they divide the number line into intervals where the expression's sign remains constant. To find these points, we set each factor equal to zero:

• $x - 3 = 0 => x = 3$
• $x^3 - 6 = 0 => x^3 = 6 => x = \sqrt[3]{6}$

So, our critical points are $x = 3$ and $x = \sqrt[3]{6}$. Since $1^3 = 1$ and $2^3 = 8$, we know that $1 < \sqrt[3]{6} < 2$. Therefore, $\sqrt[3]{6}$ is approximately 1.82.

The critical points are like the signposts on our number line, marking the potential changes in the expression's value. They are the values of x where the factors switch from positive to negative or vice-versa. These points are essential for creating a sign chart, which is a visual tool that helps us analyze the intervals where the inequality holds. By finding the critical points, we've essentially identified the boundaries of the regions where the expression might be less than zero. Understanding the location and nature of these critical points is vital for accurately determining the solution set. Each critical point represents a potential endpoint or boundary for the intervals that satisfy our inequality. Now that we have these points, we're ready to build our sign chart and analyze the behavior of the expression within each interval.

A sign chart is a powerful tool for visualizing how the sign of an expression changes across different intervals. We'll use it to determine where $(x - 3)(x^3 - 6) < 0$. First, we draw a number line and mark our critical points, $\sqrt[3]{6}$ and 3:

       --------(\sqrt[3]{6})--------(3)-------->

These critical points divide the number line into three intervals: $(-\infty, \sqrt[3]{6})$, $(\sqrt[3]{6}, 3)$, and $(3, \infty)$. Now, we'll choose a test value within each interval and evaluate the sign of each factor, $(x - 3)$ and $(x^3 - 6)$:

A sign chart is like a map that guides us through the solution landscape. It provides a clear visual representation of the expression's behavior in each interval. The chart shows how the signs of the individual factors combine to determine the sign of the overall expression. This is crucial because we're looking for the intervals where the expression is negative, as indicated by the “< 0” in our original inequality. By choosing test values and evaluating the signs, we're essentially taking samples to understand the general trend within each interval. This systematic approach ensures that we don't miss any parts of the solution set. The sign chart is a powerful aid in making the abstract concept of inequalities more concrete and understandable. It allows us to see the solution intervals at a glance, making the final step of determining the solution set much easier.

From the sign chart, we can see that $(x - 3)(x^3 - 6) < 0$ when $x$ is in the interval $(\sqrt[3]{6}, 3)$. Therefore, the solution set to the inequality $x^4 - 3x^3 < 6(x - 3)$ is:

$\sqrt[3]{6} < x < 3$

In interval notation, this is written as:

$(\sqrt[3]{6}, 3)$

This is the final piece of the puzzle! We've successfully navigated the algebraic maze and arrived at the solution set. The solution set represents all the values of x that make the original inequality true. In this case, it's all the numbers between the cube root of 6 and 3, not including the endpoints. This means that any value of x within this interval, when plugged into the original inequality, will result in a true statement. Determining the solution set is the culmination of our efforts, the moment where we can confidently say we've solved the problem. The solution set is the destination we were aiming for from the very beginning. Now that we have it, we can be sure that we've fully addressed the inequality and found all the values of x that satisfy the given conditions. This process highlights the power of algebraic methods in solving inequalities and provides a framework for tackling similar problems in the future.

So there you have it! We've successfully solved the inequality $x^4 - 3x^3 < 6(x - 3)$ using algebraic methods. We rearranged the inequality, factored the polynomial, found the critical points, created a sign chart, and finally determined the solution set: $(\sqrt[3]{6}, 3)$. This journey demonstrates the power of algebraic techniques in tackling complex problems. By breaking down the problem into smaller, manageable steps, we were able to systematically find the solution. Understanding these steps is crucial for mastering algebra and solving a wide range of mathematical problems. Remember, practice makes perfect! The more you work with inequalities and factoring, the more confident you'll become in your ability to solve them. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. Solving this type of inequality involves a combination of algebraic manipulation and logical reasoning. It is a great exercise in mathematical thinking. We hope you found this guide helpful and that you're now better equipped to tackle similar problems on your own. Keep exploring the world of math, guys! There's always something new and exciting to discover. This example showcases how a methodical approach, combined with the right algebraic tools, can lead to a successful solution. We've walked through each step carefully, emphasizing the reasoning behind each manipulation. This approach not only helps in solving this specific problem but also builds a strong foundation for tackling future challenges in mathematics. Keep honing your skills, and you'll find that the world of inequalities becomes increasingly accessible and engaging.