Solving Inequalities: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into the world of inequalities, specifically tackling the problem of $x^3 + 2x^2 ≤ 8x$. Don't worry, it might seem a bit intimidating at first, but trust me, we'll break it down into easy-to-digest steps. By the end of this, you'll be a pro at solving this type of problem. This isn't just about finding an answer; it's about understanding how to find the answer and why it works. We're going to use a methodical approach, ensuring you grasp every single step involved. So, grab your pencils and let's get started. Remember, the key to mastering math is practice, practice, practice! The more problems you solve, the more confident you'll become. We'll start with the basics, moving through the necessary algebraic manipulations and finally, to the solution set. This will involve factoring, finding critical points, and testing intervals. The aim is to ensure you not only get the right answer but also understand the reasoning behind each step, building a solid foundation in algebra. Learning inequalities can be incredibly useful in a wide array of applications, from modeling real-world situations to understanding complex mathematical concepts. So, let’s transform this seemingly complex inequality into something manageable. We'll start by rearranging the inequality so we have all terms on one side, which is a crucial first step in dealing with polynomial inequalities. Remember, moving terms around requires us to pay attention to the signs – a detail that's essential for arriving at the correct solution. Stick with me, and I'll walk you through each stage clearly. This systematic approach is designed to eliminate confusion and make the process as straightforward as possible. Ready to begin our journey to conquer this inequality? Let's go!

Step 1: Rearrange the Inequality

Alright, first things first, we want to bring everything to one side of the inequality. This is a fundamental move that sets us up for the next steps. Our original inequality is $x^3 + 2x^2 ≤ 8x$. To get all terms on one side, let's subtract $8x$ from both sides. This gives us:

$x^3 + 2x^2 - 8x ≤ 0$

See? Simple, right? This rearrangement sets the stage for us to factor the expression, which is key to finding the values of $x$ that satisfy the inequality. Always remember, when manipulating inequalities, you must preserve the relationship between the quantities. This initial step is very critical; by getting all the non-zero terms on one side, we're setting up the problem to utilize factoring techniques. Think of it as preparing your canvas before you start painting. Everything else we do will hinge on this, so it is important to pay close attention to ensure no terms are missed or incorrectly manipulated. We are essentially reorganizing the equation, setting it equal to zero, to find what are known as the critical points. These points will serve as boundaries for our final solution and allow us to identify the intervals where the inequality is true. Mastering this initial step is critical in dealing with more complex inequalities. Now that the stage is set, let’s move on to the next critical part of the process, factoring, to unravel the inequality and find our solution set. Ready to proceed? Let's dive in and see how we can simplify things even further.

Step 2: Factor the Expression

Now for the fun part: factoring! Our expression is $x^3 + 2x^2 - 8x ≤ 0$. The goal here is to break down the polynomial into simpler factors. Notice that each term has an $x$, so we can start by factoring out an $x$:

$x(x^2 + 2x - 8) ≤ 0$

Nice, that simplifies things already! Next, we need to factor the quadratic expression $x^2 + 2x - 8$. Can you think of two numbers that multiply to -8 and add up to 2? Yep, those numbers are 4 and -2. So, we can factor the quadratic as $(x + 4)(x - 2)$. Thus, the complete factored form of our expression is:

$x(x + 4)(x - 2) ≤ 0$

This factored form is incredibly useful. It shows us the roots of the equation when it equals zero (i.e., the critical points). Remember, factoring simplifies the polynomial, making it easier to identify the key points. This step transforms the complex expression into something we can analyze more easily. We are now one step closer to isolating the intervals where the inequality holds true. These critical points represent where the expression changes sign, which is fundamental to solving the inequality. Mastering factoring is a skill that will serve you well in many areas of mathematics. With the expression now factored, we can proceed to identify those crucial values of x. Let's move on to the next step, where we'll determine the critical points of the inequality.

Step 3: Find the Critical Points

The critical points are the values of $x$ that make the expression equal to zero. These are the points where the expression can change signs, and they are essential for solving the inequality. From our factored form, $x(x + 4)(x - 2) ≤ 0$, we find the critical points by setting each factor equal to zero:

• $x = 0$
• $x + 4 = 0 => x = -4$
• $x - 2 = 0 => x = 2$

So, our critical points are $x = -4, x = 0,$ and $x = 2$. These are the values that will divide our number line into intervals. These points give us specific values where the expression is exactly equal to zero, providing the necessary boundaries for determining where our inequality is true. Note that these are not the solutions themselves but serve as a framework for the next stage of our work. Plotting these points on a number line helps to visualize the different intervals where we will test the inequality. Now that we've found these critical points, the next step involves testing intervals to determine where the inequality holds. Understanding the critical points is key to unlocking the final solution to the inequality, which leads us to testing our intervals.

Step 4: Test the Intervals

Okay, here's where we figure out which intervals satisfy our inequality. We have the critical points -4, 0, and 2, which divide the number line into four intervals: $(-\\,\infty, -4)$, $(-4, 0)$, $(0, 2)$, and $(2, +\infty)$. We need to pick a test value within each interval and substitute it into our factored inequality, $x(x + 4)(x - 2) ≤ 0$, to see if it makes the inequality true. Let's do this systematically:

1. Interval $(-\\,\infty, -4)$: Choose $x = -5$. Substitute into the inequality: $-5(-5 + 4)(-5 - 2) = -5(-1)(-7) = -35$. Since $-35 ≤ 0$, this interval satisfies the inequality.
2. Interval $(-4, 0)$: Choose $x = -1$. Substitute into the inequality: $-1(-1 + 4)(-1 - 2) = -1(3)(-3) = 9$. Since $9 > 0$, this interval does not satisfy the inequality.
3. Interval $(0, 2)$: Choose $x = 1$. Substitute into the inequality: $1(1 + 4)(1 - 2) = 1(5)(-1) = -5$. Since $-5 ≤ 0$, this interval satisfies the inequality.
4. Interval $(2, +\infty)$: Choose $x = 3$. Substitute into the inequality: $3(3 + 4)(3 - 2) = 3(7)(1) = 21$. Since $21 > 0$, this interval does not satisfy the inequality.

Now, we know which intervals satisfy the inequality. Testing intervals is a crucial method to solve inequalities, as it provides a way to verify the intervals where the original condition is met. This technique helps determine which parts of the number line are part of the solution set. During the interval testing, we're basically looking for the segments of the number line that provide us with a result that satisfies the inequality. Remember that our intervals are defined by the critical points, and each interval is evaluated independently by testing a value within each. By substituting a test value in each interval, we ascertain whether the inequality is true or false for that interval. Next, we put it all together to state our final solution. Let’s identify our final solution set!

Step 5: Write the Solution

Alright, we've tested our intervals, and now we need to put it all together to write the solution. Remember that our inequality is $x(x + 4)(x - 2) ≤ 0$. We found that the intervals $(-\\,\infty, -4)$ and $(0, 2)$ satisfy the inequality. Also, the critical points themselves, $-4$, $0$, and $2$, also make the expression equal to zero, which satisfies the