Solving Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of inequalities, specifically tackling the algebraic solution of a rational inequality. We will break down how to solve the inequality: $\frac{(6-x)^3(8x+5)}{x^3-27}<0$. Inequalities might seem intimidating at first, but trust me, with a systematic approach, you'll be solving them like a pro in no time. We'll cover everything from identifying critical points to using sign charts, making sure you grasp each concept along the way. So, let's get started and unravel this mathematical puzzle together!

Understanding Inequalities

Before we jump into the specifics of this inequality, let's make sure we're all on the same page about what inequalities are and why they're important. Inequalities, unlike equations, don't give us a single solution. Instead, they define a range of values that satisfy a given condition. Think of it like this: instead of finding the exact value of x that makes an equation true, we're finding all the values of x that make an expression greater than, less than, greater than or equal to, or less than or equal to a certain value.

Why are Inequalities Important?

You might be wondering, "Why should I care about inequalities?" Well, they pop up everywhere in real-world applications! From optimizing business costs to modeling physical constraints in engineering, inequalities are essential tools. For instance, consider a scenario where a company wants to maximize profit. They need to ensure that their production costs are less than their revenue. This directly translates into an inequality. Similarly, in physics, you might encounter inequalities when dealing with the range of projectile motion or the stability of structures. Understanding inequalities allows us to make informed decisions and solve practical problems across various fields. Inequalities help define boundaries and constraints, allowing us to work within realistic parameters. They provide a more nuanced understanding of the world around us, where conditions aren't always exact but fall within a certain range.

Key Concepts in Solving Inequalities

When solving inequalities, there are a few key concepts to keep in mind. First, we often deal with intervals of solutions rather than single values. These intervals can be open (excluding the endpoints), closed (including the endpoints), or a combination of both. We use interval notation to represent these solutions, which might look like (-∞, 5), [2, 7], or (3, ∞). Second, the direction of the inequality matters. Multiplying or dividing by a negative number flips the inequality sign. This is a crucial rule to remember to avoid common mistakes. Third, we often use sign charts to analyze the behavior of expressions over different intervals. Sign charts help us visualize where an expression is positive, negative, or zero, which is essential for solving rational and polynomial inequalities. Keeping these concepts in mind will make the process of solving inequalities much smoother and more intuitive. Understanding these fundamentals is like having the right tools in your toolbox – they enable you to tackle any inequality problem with confidence.

Step-by-Step Solution

Okay, let's dive into the heart of the matter: solving the inequality $\frac{(6-x)^3(8x+5)}{x^3-27}<0$. We'll break this down into manageable steps so it's super clear. Our goal is to find all the values of x that make this expression less than zero. Buckle up, guys, we're about to tackle this step by step!

1. Identify Critical Points

The first thing we need to do is find the critical points. These are the values of x that make either the numerator or the denominator of our rational expression equal to zero. Why are critical points important? Because they divide the number line into intervals where the expression's sign (positive or negative) remains constant. In other words, the expression can only change signs at these critical points. So, let's find them.

Numerator

The numerator is $(6-x)^3(8x+5)$. To find its zeros, we set each factor equal to zero:

• $(6-x)^3 = 0$ => $6-x = 0$ => $x = 6$
• $8x+5 = 0$ => $8x = -5$ => $x = -\frac{5}{8}$

So, the critical points from the numerator are $x = 6$ and $x = -\frac{5}{8}$.

Denominator

The denominator is $x^3 - 27$. We need to solve $x^3 - 27 = 0$. This looks like a difference of cubes, which we can factor as:

• $x^3 - 27 = (x-3)(x^2 + 3x + 9)$

Setting each factor equal to zero:

• $x-3 = 0$ => $x = 3$
• $x^2 + 3x + 9 = 0$

For the quadratic factor, let's check its discriminant ($b^2 - 4ac$) to see if it has real roots:

• $D = 3^2 - 4(1)(9) = 9 - 36 = -27$

Since the discriminant is negative, the quadratic factor has no real roots. Thus, the only critical point from the denominator is $x = 3$.

Critical Points Summary: Our critical points are $x = -\frac{5}{8}$, $x = 3$, and $x = 6$. These points are like the signposts on our number line, guiding us to the intervals where our inequality's sign stays consistent.

2. Create a Sign Chart

Now that we have our critical points, we're going to create a sign chart. This chart helps us visualize the sign of our expression, $\frac{(6-x)^3(8x+5)}{x^3-27}$, in each interval defined by the critical points. Trust me, a sign chart is your best friend when solving inequalities!

Setting up the Chart

1. Draw a number line. Mark our critical points, $-0.625$, $3$, and $6$, on the line. These points divide the number line into four intervals: $(-\infty, -\frac{5}{8})$, $(-\frac{5}{8}, 3)$, $(3, 6)$, and $(6, \infty)$.
2. List each factor from our inequality: $(6-x)^3$, $(8x+5)$, and $(x^3-27)$ [which we can think of as $(x-3)(x^2+3x+9)$ but since $x^2+3x+9$ is always positive, we only really need to track $(x-3)$].
3. Create a table with these factors as rows and the intervals as columns. This is where the magic happens!

Filling in the Chart

For each interval, we'll pick a test value and plug it into each factor to determine its sign (positive or negative) in that interval. Remember, the sign of the factor will be consistent within each interval.

1. Interval $(-\infty, -\frac{5}{8})$: Let's pick $x = -1$ as our test value.
   • $(6 - (-1))^3 = 7^3 > 0$ (Positive)
   • $8(-1) + 5 = -3 < 0$ (Negative)
   • $(-1)^3 - 27 = -28 < 0$ (Negative)
2. Interval $(-\frac{5}{8}, 3)$: Let's pick $x = 0$ as our test value.
   • $(6 - 0)^3 = 6^3 > 0$ (Positive)
   • $8(0) + 5 = 5 > 0$ (Positive)
   • $0^3 - 27 = -27 < 0$ (Negative)
3. Interval $(3, 6)$: Let's pick $x = 4$ as our test value.
   • $(6 - 4)^3 = 2^3 > 0$ (Positive)
   • $8(4) + 5 = 37 > 0$ (Positive)
   • $4^3 - 27 = 37 > 0$ (Positive)
4. Interval $(6, \infty)$: Let's pick $x = 7$ as our test value.
   • $(6 - 7)^3 = (-1)^3 < 0$ (Negative)
   • $8(7) + 5 = 61 > 0$ (Positive)
   • $7^3 - 27 = 316 > 0$ (Positive)

Determining the Expression's Sign

Now, to find the sign of the entire expression in each interval, we multiply the signs of the factors. Remember, a negative times a negative is a positive, and so on.

• $(-\infty, -\frac{5}{8})$: $(+)(-) / (-) = (+)$
• $(-\frac{5}{8}, 3)$: $(+)(+) / (-) = (-)$
• $(3, 6)$: $(+)(+) / (+) = (+)$
• $(6, \infty)$: $(-)(+) / (+) = (-)$

Our sign chart now clearly shows where the expression is positive or negative. This is a crucial step in solving the inequality!

3. Determine the Solution Set

We're in the home stretch now! Remember, we want to find the intervals where $\frac{(6-x)^3(8x+5)}{x^3-27}<0$, meaning we're looking for the intervals where the expression is negative.

From our sign chart, we see that the expression is negative in the intervals $(-\frac{5}{8}, 3)$ and $(6, \infty)$. But there's one more thing we need to consider: the endpoints.

Endpoint Considerations

• At $x = -\frac{5}{8}$, the numerator is zero, making the entire expression zero. Since we want the expression to be less than zero, we exclude this point. So, we use a parenthesis.
• At $x = 3$, the denominator is zero, making the expression undefined. We always exclude points that make the denominator zero. So, we use a parenthesis.
• At $x = 6$, the numerator is zero, making the entire expression zero. Again, since we want the expression to be less than zero, we exclude this point. So, we use a parenthesis.

The Solution

Putting it all together, the solution to the inequality $\frac{(6-x)^3(8x+5)}{x^3-27}<0$ is:

• $\left(-\frac{5}{8}, 3\right) \cup (6, \infty)$

This is our final answer! We've successfully navigated through the critical points, the sign chart, and the endpoint considerations to find the solution set. We've nailed it!

Common Mistakes to Avoid

Solving inequalities can be tricky, and it's easy to make small mistakes that lead to incorrect answers. Let's talk about some common pitfalls and how to steer clear of them. Knowing these common mistakes can save you a lot of headaches!

1. Forgetting to Flip the Inequality Sign

One of the most common errors is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Remember, this is a crucial rule! If you multiply or divide by a negative, you've got to flip that sign. For example, if you have $-2x < 4$, dividing by -2 gives you $x > -2$, not $x < -2$. Always double-check this step to avoid this frequent mistake.

2. Ignoring Critical Points from the Denominator

It's essential to consider the critical points that make the denominator zero. These points are not part of the solution because they make the expression undefined. Forgetting these points can lead to including values in your solution set that shouldn't be there. Always remember to identify and exclude these critical points from your final answer.

3. Incorrectly Interpreting the Sign Chart

A sign chart is a powerful tool, but it's only as good as your interpretation. Make sure you correctly identify the intervals where the expression satisfies the inequality. If you're looking for where the expression is less than zero, focus on the negative intervals. Double-check your signs and ensure you're picking the right intervals. A small error in interpreting the chart can completely change your solution.

4. Including Endpoints Incorrectly

Deciding whether to include or exclude endpoints can be tricky. Remember, if the inequality is strict (i.e., < or >), you exclude the endpoints. If it's non-strict (≤ or ≥), you include them, unless they make the denominator zero. Always consider the inequality symbol and whether the endpoints make the expression zero or undefined. This careful consideration will ensure you get the endpoints right.

5. Not Checking Your Solution

Finally, one of the best ways to avoid mistakes is to check your solution. Pick a test value from each interval in your solution set and plug it back into the original inequality. If the inequality holds true, you're on the right track. If not, there's likely a mistake somewhere. Checking your solution is like having a safety net – it catches errors before they become a problem.

Conclusion

And there you have it! We've walked through the process of solving the inequality $\frac{(6-x)^3(8x+5)}{x^3-27}<0$ step by step. We covered everything from finding critical points and creating sign charts to avoiding common mistakes. Inequalities might seem daunting, but with a systematic approach and a clear understanding of the key concepts, you can tackle them with confidence. Remember, practice makes perfect, so keep solving those inequalities! You've got this, guys!

By mastering these techniques, you'll be well-equipped to handle a wide range of inequality problems. Keep practicing, stay patient, and remember to break down each problem into manageable steps. Happy solving!