Solving Absolute Value Inequalities: Finding The Values Of X

Hey math enthusiasts! Today, we're diving into a cool problem involving absolute values. Specifically, we're going to figure out the values of x that satisfy the inequality: $|x^2 - 9| ≤ 0$. Don't worry, it might look a bit intimidating at first, but we'll break it down step by step to make it super clear. This is a classic example of how understanding absolute value can unlock solutions to some interesting mathematical puzzles, and it's a fundamental concept in algebra that helps you build a solid foundation for more complex topics. So, let's get started, shall we?

Understanding the Absolute Value

First off, let's make sure we're all on the same page about what absolute value means. The absolute value of a number is its distance from zero on the number line. It's always a non-negative value. Think of it like this: regardless of whether a number is positive or negative, its absolute value strips away the sign and tells you how far away it is from zero. For example, the absolute value of 5, denoted as |5|, is 5. And the absolute value of -5, denoted as |-5|, is also 5. So, absolute values are always greater than or equal to zero. This is a crucial point because it significantly limits the possibilities in our inequality. We need to remember this property because it's the key to understanding the inequality we're dealing with. Knowing that absolute values are always non-negative is the cornerstone of solving the problem. So, always remember that the absolute value of anything will always be zero or positive. Now, let’s move on to the actual problem.

Now, let’s bring this understanding back to our inequality, $|x^2 - 9| ≤ 0$. What this tells us is that the absolute value of $(x^2 - 9)$ must be less than or equal to zero. But wait a minute! We've just learned that absolute values can never be negative. They are always either positive or zero. Thus, the only way for the absolute value of an expression to be less than or equal to zero is if that expression itself equals zero. In the context of our problem, that means $|x^2 - 9| = 0$. This is the critical insight that simplifies the whole problem. We are no longer dealing with an inequality; instead, we can directly solve for when the expression inside the absolute value signs equals zero. The key is recognizing that the only possible solution is where the absolute value equals zero because it cannot be negative. This simplifies the approach significantly and allows us to get closer to the solution.

Solving the Inequality

Okay, folks, based on what we’ve talked about, to solve $|x^2 - 9| ≤ 0$, we really need to solve for when the inside of the absolute value is equal to zero. In other words, we need to find the values of x such that $x^2 - 9 = 0$. This is a straightforward quadratic equation. We can solve this in a couple of ways: by factoring or by isolating x. Let's go with factoring first, because it's usually the quickest method. We're looking for two numbers that multiply to give -9 and add up to give 0. Those numbers are 3 and -3. So, we can factor the equation as $(x - 3)(x + 3) = 0$.

Now, for this equation to be true, either $(x - 3) = 0$ or $(x + 3) = 0$. Solving each of these gives us two possible solutions. If $x - 3 = 0$, then $x = 3$. If $x + 3 = 0$, then $x = -3$. This tells us that the values of x that make $x^2 - 9 = 0$ are x = 3 and x = -3. Now, we want to double-check that these values work in our original inequality, $|x^2 - 9| ≤ 0$. For x = 3, we have $|3^2 - 9| = |9 - 9| = |0| = 0$, which indeed satisfies the inequality. For x = -3, we have $|(-3)^2 - 9| = |9 - 9| = |0| = 0$, which also satisfies the inequality. Thus, both -3 and 3 are valid solutions, confirming the previous step's accuracy.

Alternatively, we could have used the square root method to solve $x^2 - 9 = 0$. You could rearrange the equation to isolate $x^2$, giving us $x^2 = 9$. Then, taking the square root of both sides, we get $x = ±√9$, which means $x = ±3$. Regardless of the method you choose, the solutions will be the same. The important thing is that you understand the process and can reliably solve these types of equations. Therefore, regardless of how you tackle it, whether through factoring or isolation, both approaches lead to the same solutions: -3 and 3. And this reinforces the power of mathematics: the same problem, different roads, but always the same correct answer.

Choosing the Correct Answer

Alright, we have successfully solved the problem and found the values of x that satisfy the inequality. Now let's examine the options and select the correct one. The question is: $|x^2 - 9| ≤ 0$. We've determined that the only values of x that work are -3 and 3. Let's compare this with the provided options:

A. {-3, 3}: This matches our solution perfectly. These are the two values of x we found that satisfy the original inequality. It is the correct answer. B. -3 < x < 3: This represents all the numbers between -3 and 3, but not including -3 and 3. This is incorrect since we need x to be specifically -3 and 3. C. -3 ≤ x ≤ 3: This represents all numbers between -3 and 3, including -3 and 3. While this includes the values we found, it also includes many other values that do not satisfy the original inequality. D. x > 3: This option indicates all the numbers greater than 3, which is not correct since -3 and 3 are the only possible solutions. E. x < -3: This indicates all numbers less than -3, so this is also incorrect.

So, by carefully analyzing each option, we can confidently say that option A, {-3, 3}, is the correct answer. It perfectly describes the values of x that make the absolute value expression less than or equal to zero. This concludes our exploration of this absolute value inequality. Understanding these concepts helps you grasp the foundational principles that will be used in a variety of mathematical topics. It is a win-win because it builds your confidence in problem-solving and also boosts your understanding of how mathematics works.