Solving Inequalities: A Sign Chart Guide

Hey math enthusiasts! Today, we're diving deep into the world of inequalities and how to solve them using a super helpful tool called a sign chart. We'll use the example of the expression $\frac{8x^3}{x-7} \leq 0$ to guide us. Sign charts are a visual and organized way to determine the intervals where an expression is positive, negative, or zero. This is super important when we need to find the solutions to inequalities. Understanding how to create and interpret sign charts is key to mastering algebra and calculus. Let's break it down step by step, making sure we get a solid grasp of the concepts. This method is incredibly useful not just for this specific problem, but for a whole range of inequality problems you might encounter in the future. Ready to get started? Let’s jump in!

Understanding the Basics of Sign Charts

Alright guys, before we tackle our specific inequality, let's get familiar with what a sign chart actually is. A sign chart, at its core, is a number line that helps us visualize the sign (positive or negative) of an expression across different intervals. The chart is divided by critical points – these are the values of x where the expression either equals zero or is undefined. In our example, $\frac{8x^3}{x-7} \leq 0$, we need to find where the numerator and denominator could cause the expression to change its sign. The main idea behind a sign chart is to divide the number line into intervals. Then, we choose a test value within each interval and substitute it into the expression. The sign of the result tells us whether the expression is positive or negative in that interval. This method helps us avoid the need to analyze the expression's behavior in each interval. This makes the solving process more efficient and reduces the chance of making errors. Using sign charts provides a clear, organized approach to solving inequalities. Now, let's prepare the groundwork for solving our specific problem.

First, consider the numerator, $8x^3$. This expression equals zero when $x = 0$. Next, consider the denominator, $x - 7$. This expression equals zero when $x = 7$. Also, it’s important to note that the denominator cannot be zero, as it would make the entire expression undefined. So, our critical points are $x = 0$ and $x = 7$. These are the points where the expression can change signs. These points divide the number line into three intervals: $(-\infty, 0)$, $(0, 7)$, and $(7, \infty)$. By analyzing the sign of the expression in each of these intervals, we will be able to determine the solution to the inequality.

Constructing the Sign Chart for $\frac{8x^3}{x-7} \leq 0$

Okay, let's get down to business and build our sign chart for $\frac{8x^3}{x-7} \leq 0$. Remember, we've identified our critical points as $x = 0$ and $x = 7$. Now, draw a number line and mark these points. Be sure to use open circles at values of $x$ where the function is undefined to reflect this fact in your sign chart. We are not including 7 because the function is undefined at this point. This means that 7 is not part of the solution. Then, we are going to divide the number line into three intervals using these points: $(-\infty, 0)$, $(0, 7)$, and $(7, \infty)$. Next, we will pick a test value in each interval. This value will be used to determine the sign of the entire expression in that interval. For the interval $(-\infty, 0)$, let’s choose $x = -1$. Substituting $x = -1$ into the expression gives us $\frac{8(-1)^3}{(-1)-7} = \frac{-8}{-8} = 1$. The result is positive, so the sign in the interval $(-\infty, 0)$ is positive (+). For the interval $(0, 7)$, let’s pick $x = 1$. When we plug it into the expression, we get $\frac{8(1)^3}{(1)-7} = \frac{8}{-6} = -\frac{4}{3}$. The result is negative, so the sign in the interval $(0, 7)$ is negative (-). For the interval $(7, \infty)$, let’s choose $x = 8$. Substituting $x = 8$, we get $\frac{8(8)^3}{(8)-7} = \frac{4096}{1} = 4096$. The result is positive, so the sign in the interval $(7, \infty)$ is positive (+). We've successfully built our sign chart!

Now, here’s a tip: in a sign chart, the factors that appear with odd powers will change the sign around the critical points. For instance, in our expression, the factor x appears to the power of 3 (which is an odd power). This results in a sign change when crossing through the critical point at x = 0. The factor (x-7) appears to the power of 1 (also an odd power), which means the sign changes when crossing through the critical point at x = 7. If the power were even, the sign wouldn't change. Also, don't forget to take into account the constant factors, such as 8 in our expression, which are positive and don't affect the sign in our sign chart.

Interpreting the Sign Chart and Solving the Inequality

Alright, so we have our sign chart. Now, what does it all mean? The sign chart displays the intervals where our expression, $\frac{8x^3}{x-7}$, is positive or negative. The question is, where is $\frac{8x^3}{x-7} \leq 0$? We are looking for the intervals where the expression is less than or equal to zero. This means we are looking for negative values (the intervals marked with a “-” sign) and where the expression is equal to zero. From our sign chart, the expression is negative in the interval $(0, 7)$. At x = 0, the expression is equal to zero (since the numerator $8x^3$ is zero). The expression is undefined at x = 7, so we do not include it. Thus, our solution includes the interval $(0, 7)$. Since the inequality includes