Solving Inequalities Algebraically: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of algebraic inequalities. Specifically, we're going to solve the inequality algebraically: $\frac{(x-4)^2}{x^2-9} \geq 0$. Don't worry, it might look a little intimidating at first, but trust me, we'll break it down into manageable steps. By the end of this guide, you'll be able to tackle similar problems with confidence. So, let's get started, shall we?

Understanding the Basics of Algebraic Inequalities

Before we jump into the problem, let's quickly recap what algebraic inequalities are all about. An inequality is a mathematical statement that compares two expressions using inequality symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving an inequality means finding the range of values that make the statement true. Unlike equations, which usually have a single solution or a few discrete solutions, inequalities often have a range of solutions, represented by intervals on the number line. Understanding this is super important because it forms the backbone of how we approach our problem. We're looking for the values of 'x' that satisfy the given condition, and we'll express our answer as intervals where the inequality holds true. These intervals will be determined by the critical points where the expression changes its sign, which are the real zeros (roots) of the numerator and denominator.

Key Concepts and Terminology

To make sure we're all on the same page, let's define a few key terms:

• Inequality: A mathematical statement comparing two expressions using inequality symbols.
• Solution: The set of values that satisfy the inequality.
• Interval: A range of values on the number line, often used to represent the solution set.
• Zeros: The values of x that make an expression equal to zero. These are crucial because they're the points where the expression can change signs.
• Numerator: The top part of a fraction.
• Denominator: The bottom part of a fraction.

Now, armed with these concepts, we're ready to tackle our inequality. Remember, our goal is not just to find a solution but to find the complete solution set, i.e., all values of 'x' for which the inequality holds true. This involves careful consideration of the zeros of the numerator and denominator, which will help us identify the intervals where the expression is positive, negative, or equal to zero. It's like a treasure hunt, and we're looking for all the hidden spots where the inequality's true.

Step-by-Step Solution of the Inequality

Alright, let's get down to business. We'll break down the inequality $\frac{(x-4)^2}{x^2-9} \geq 0$ step-by-step. This is the fun part where we get to apply our knowledge and skills to find the solution. Each step is designed to simplify the problem and bring us closer to the answer. So, buckle up, and let's go! Remember, the goal is to find all the values of 'x' that satisfy the inequality.

Step 1: Define f(x)

First, we need to define the left side of the inequality as f(x). So, let: $f(x) = \frac{(x-4)^2}{x^2-9}$. This makes it easier to refer to the expression and work with it. Defining f(x) is simply a notational convenience that helps us keep track of what we're working with. It's like giving our expression a name so we can refer to it easily throughout the process.

Step 2: Find the Zeros of f(x) (Numerator and Denominator)

Next, we need to find the real zeros of f(x). These are the values of 'x' that make either the numerator or the denominator equal to zero.

• Numerator: $(x-4)^2 = 0$. This gives us $x = 4$. Notice that since the numerator is squared, x = 4 is a zero of even multiplicity.
• Denominator: $x^2 - 9 = 0$. This factors into $(x-3)(x+3) = 0$. This gives us $x = 3$ and $x = -3$. The values x = 3 and x = -3 are not included in the solution because they would make the denominator zero, which is undefined. These are the critical points where the expression can change sign.

Step 3: Identify the Intervals

Now, we use the zeros we found to divide the number line into intervals. The zeros are -3, 3, and 4. This gives us the following intervals:

• $(-\infty, -3)$
• $(-3, 3)$
• $(3, 4)$
• $(4, \infty)$

These intervals are crucial because they represent the possible regions where the inequality might be true or false. In each interval, the expression $f(x)$ will have a consistent sign (either positive or negative), because the expression can only change its sign at the zeros.

Step 4: Test Intervals

We need to test each interval to see where $f(x) \geq 0$. We'll pick a test value within each interval and substitute it into $f(x)$ to see if the result is positive or negative. This is a very important step to ensure the integrity of the solution.

• Interval $(-\infty, -3)$: Let's pick $x = -4$. Then, $f(-4) = \frac{(-4-4)^2}{(-4)^2-9} = \frac{64}{7} > 0$. So, this interval is part of our solution.
• Interval $(-3, 3)$: Let's pick $x = 0$. Then, $f(0) = \frac{(0-4)^2}{(0)^2-9} = \frac{16}{-9} < 0$. This interval is not part of our solution.
• Interval $(3, 4)$: Let's pick $x = 3.5$. Then, $f(3.5) = \frac{(3.5-4)^2}{(3.5)^2-9} = \frac{0.25}{3.25} > 0$. So, this interval is part of our solution.
• Interval $(4, \infty)$: Let's pick $x = 5$. Then, $f(5) = \frac{(5-4)^2}{(5)^2-9} = \frac{1}{16} > 0$. So, this interval is part of our solution.

Step 5: Consider the Zeros and Write the Solution

• x = 4: Since $x = 4$ makes the numerator zero, $f(4) = 0$, which satisfies $f(x) \geq 0$. So, we include $x = 4$ in our solution.
• x = -3 and x = 3: These values make the denominator zero, so they are not included in the solution.

Therefore, the solution to the inequality is $(-\infty, -3) \cup (3, \infty) \cup \{4\}$.

Graphing the Solution

To better visualize the solution, we can graph it on a number line. The number line will have open circles at -3 and 3 (because these values are not included) and a closed circle at 4 (because it is included). The solution set consists of the intervals where the graph is above or on the x-axis. Graphing the solution provides a visual representation that complements our algebraic steps, making it easier to understand the overall picture. This visual aid reinforces the intervals where the inequality holds and clarifies the role of critical points and zeros.

Conclusion: Mastering Algebraic Inequalities

And there you have it! We've successfully solved the inequality $\frac{(x-4)^2}{x^2-9} \geq 0$. This process helps you to master algebraic inequalities and equips you with the tools to solve a wide variety of similar problems. Remember, practice is key, and the more you practice, the more comfortable and confident you'll become. Keep at it, guys, and you'll be inequality-solving pros in no time! Always remember to carefully consider the zeros of the numerator and denominator, test the intervals, and express your solution accurately. Keep practicing, and you'll become a pro at this. Happy solving, and keep exploring the fascinating world of mathematics!