Inequality Solution: $3(x+9)^{\frac{3}{4}} \leq 24$

Welcome to our deep dive into solving inequalities, a fundamental skill in mathematics! Today, we're going to tackle a specific problem: $3(x+9)^{\frac{3}{4}} \leq 24$. Inequalities, unlike equations, don't just give us a single answer; they often represent a range of possible values that satisfy the given condition. This makes them incredibly useful for describing real-world scenarios where boundaries and limits are involved. Think about it – when do you not have a strict limit? Almost always! Whether it's a budget, a speed limit, or even the amount of time you have to complete a task, inequalities are at play. In this article, we'll break down the process of solving this particular inequality step-by-step, ensuring you not only get the answer but also understand the reasoning behind each move. We'll cover everything from isolating the variable to considering the implications of fractional exponents. So, grab your thinking caps, and let's get ready to unravel the mystery behind this mathematical expression. Understanding how to manipulate these forms is key to building a strong foundation in algebra and beyond. It's not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving that can be applied to countless other challenges, both within and outside the realm of mathematics. We'll make sure to explain the 'why' behind each step, transforming this potentially daunting problem into a clear and manageable process. Remember, practice is crucial, and by working through this example, you're taking a significant step towards mastering inequalities.

Understanding the Components of the Inequality

Before we dive into solving, let's take a moment to appreciate the different parts of our inequality: $3(x+9)^{\frac{3}{4}} \leq 24$. At its core, this is an inequality because of the 'less than or equal to' symbol ($\leq$). This symbol tells us that the expression on the left side must be smaller than or equal to the value on the right side. The left side, $3(x+9)^{\frac{3}{4}}$, is where the action is. We have a coefficient, $3$, multiplying a term with a variable, $x$. The term $(x+9)$ is raised to the power of $\frac{3}{4}$. This fractional exponent is particularly interesting. It signifies a combination of a root and a power. Specifically, $(x+9)^{\frac{3}{4}}$ means the fourth root of $(x+9)$ raised to the third power, or equivalently, the cube of the fourth root of $(x+9)$. Understanding how to handle these fractional exponents is crucial for simplifying the expression. The number $24$ on the right side is our upper bound. Our goal is to find all possible values of $x$ that keep the left side at or below this value. The parentheses around $(x+9)$ are also important; they indicate that the exponent applies to the entire expression within, not just to $x$ or $9$ individually. This order of operations is vital. When solving, we'll need to deal with the coefficient, the exponent, and the constant term inside the parentheses, carefully peeling back the layers to isolate $x$. The nature of inequalities means we must also be mindful of operations that could flip the inequality sign, although in this particular case, we'll see that most steps involve operations that preserve the direction of the inequality. This initial breakdown helps us recognize the structure of the problem and anticipate the strategies we'll need to employ for a successful solution. It's like looking at a complex machine and identifying its different parts before attempting to fix it.

Step-by-Step Solution Process

Now, let's embark on the journey to solve the inequality $3(x+9)^{\frac{3}{4}} \leq 24$. Our primary objective is to isolate the variable $x$. We'll do this by applying inverse operations in a strategic order.

Step 1: Isolate the term with the exponent.

First, we need to get the term $(x+9)^{\frac{3}{4}}$ by itself. Currently, it's being multiplied by $3$. The inverse operation of multiplication is division. So, we'll divide both sides of the inequality by $3$. Remember, dividing by a positive number does not change the direction of the inequality sign.

$\frac{3(x+9)^{\frac{3}{4}}}{3} \leq \frac{24}{3}$

This simplifies to:

$(x+9)^{\frac{3}{4}} \leq 8$

Step 2: Eliminate the fractional exponent.

This is often the trickiest part. We have $(x+9)$ raised to the power of $\frac{3}{4}$. To undo this, we need to raise both sides of the inequality to the reciprocal power, which is $\frac{4}{3}$. When we raise a power to another power, we multiply the exponents. So, $(\frac{3}{4}) \times (\frac{4}{3}) = 1$. This will leave us with just $(x+9)$ on the left side.

It's crucial to remember that when dealing with fractional exponents, especially those involving even roots (like the fourth root in $\frac{3}{4}$), we need to be cautious about negative numbers. However, since the result of $(x+9)^{\frac{3}{4}}$ is less than or equal to $8$ (a positive number), we don't need to worry about taking an even root of a negative number in this specific scenario.

Let's raise both sides to the power of $\frac{4}{3}$:

$((x+9)^{\frac{3}{4}})^{\frac{4}{3}} \leq 8^{\frac{4}{3}}$

This simplifies to:

$x+9 \leq 8^{\frac{4}{3}}$

Now, we need to evaluate $8^{\frac{4}{3}}$. Remember, this means the cube root of $8$, raised to the fourth power. The cube root of $8$ is $2$ (since $2 \times 2 \times 2 = 8$). So, $8^{\frac{4}{3}} = (\sqrt[3]{8})^4 = (2)^4$.

Calculating $2^4$: $2 \times 2 \times 2 \times 2 = 16$.

So, the inequality becomes:

$x+9 \leq 16$

Step 3: Isolate the variable $x$.

We are almost there! We have $x+9 \leq 16$. To get $x$ by itself, we need to subtract $9$ from both sides of the inequality.

$x+9 - 9 \leq 16 - 9$

This gives us:

$x \leq 7$

So, the solution to the inequality $3(x+9)^{\frac{3}{4}} \leq 24$ is $x \leq 7$. This means any number less than or equal to $7$ will satisfy the original inequality.

Considering Domain Restrictions

When dealing with fractional exponents, especially those with even denominators like our $\frac{3}{4}$, it's essential to consider the domain of the expression. The term $(x+9)^{\frac{3}{4}}$ involves a fourth root (since the denominator of the exponent is $4$). The fourth root of a number is only defined for non-negative real numbers. Therefore, the expression inside the root, $(x+9)$, must be greater than or equal to zero.

This gives us a domain restriction:

$x+9 \geq 0$

To find the values of $x$ that satisfy this, we subtract $9$ from both sides:

$x \geq -9$

This means that our solution $x \leq 7$ is only valid for values of $x$ that are also greater than or equal to $-9$. Combining these two conditions, the complete solution set for the inequality is $-9 \leq x \leq 7$. Any value of $x$ within this closed interval will satisfy the original inequality.

It's important to note that if the result on the right side of the inequality had been negative after isolating the exponent term, we would have had to consider that case separately. For instance, if we had $(x+9)^{\frac{3}{4}} \leq -5$, since a fractional exponent with an even root cannot produce a negative real number, there would be no real solution. However, in our case, since the right side ($8$) was positive, we proceeded with finding the valid range.

This consideration of domain ensures that we are only working with real numbers and that our solution is mathematically sound. It's a crucial step that distinguishes a complete solution from a partial one, especially when roots are involved. For any mathematical problem, especially those involving functions or expressions with potential restrictions, always identify and consider the domain first. This proactive approach can save a lot of trouble and lead to a more accurate and comprehensive understanding of the problem.

Conclusion: The Power of Inequalities

We have successfully navigated the complexities of solving the inequality $3(x+9)^{\frac{3}{4}} \leq 24$, arriving at the solution $-9 \leq x \leq 7$. This process has highlighted several key mathematical concepts, including the manipulation of fractional exponents, the careful application of inverse operations, and the critical importance of considering domain restrictions. Inequalities are more than just a variation of equations; they represent a spectrum of possibilities, making them invaluable tools for modeling real-world situations where boundaries and ranges are prevalent. Whether you're calculating the feasibility of a project within a budget, determining the safe operating parameters for machinery, or understanding the possible outcomes of a scientific experiment, inequalities provide the mathematical framework.

Mastering the techniques we've used today—isolating terms, raising to reciprocal powers, and respecting domain constraints—equips you with a powerful problem-solving toolkit. Remember that the fractional exponent $\frac{3}{4}$ means taking the fourth root and cubing the result, and that when we encountered this, we raised both sides to the power of $\frac{4}{3}$ to simplify. The domain restriction $x \geq -9$ arose from the fact that the fourth root must operate on a non-negative number. By combining this with our derived inequality $x \leq 7$, we found the complete range of solutions.

This journey through solving inequalities is a testament to the elegance and logic of mathematics. Each step, when understood and applied correctly, builds towards a clear and definitive answer. Keep practicing these concepts, and you'll find yourself increasingly comfortable tackling even more challenging mathematical problems. The world of mathematics is vast and fascinating, and a solid understanding of inequalities is a key that unlocks many doors.

For further exploration and to deepen your understanding of algebraic concepts and inequalities, you might find the resources at Khan Academy incredibly helpful. They offer a wealth of free lessons and practice exercises on a wide range of mathematical topics. Additionally, exploring resources from The Art of Problem Solving can provide more advanced strategies and challenging problems to hone your skills.