Solving √√x²-16 ≤ -5: A Step-by-Step Guide
Hey guys! Today, we're diving into a fun little math problem: solving the inequality √√x²-16 ≤ -5. Now, at first glance, this might seem a bit intimidating with all those square roots, but don't worry, we'll break it down step by step. We're going to make sure you not only understand the solution but also the reasoning behind each step. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into the solution, let's make sure we really understand what the problem is asking. We have an inequality, which means we're looking for a range of values for 'x' that make the statement √√x²-16 less than or equal to -5. The key here is to remember what square roots actually mean.
The square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. A crucial point to remember is that the principal square root (the one we usually deal with) is always non-negative. It can be zero, but it can never be a negative number. This little fact is going to be super important in solving our inequality.
So, with that in mind, let's think about what we have: √√x²-16. We're taking the square root of something, and then we're taking the square root of that result again. And we're saying this whole thing is less than or equal to -5. Hmmm...
This is where our understanding of square roots comes in. Can a square root ever be negative? Remember, the answer is no! This should already be giving you a hint about the solution to this inequality.
Breaking Down the Inequality
Okay, let's break this inequality down into smaller, more manageable parts. Our inequality is: √√x²-16 ≤ -5
First, let's focus on the left side of the inequality: √√x²-16. As we discussed earlier, the square root of any real number (when we're talking about principal square roots) will always be greater than or equal to zero. It can never be a negative number. This is a fundamental property of square roots that we need to keep in mind.
Now, let's consider the right side of the inequality: -5. This is a negative number. So, the inequality is essentially saying that a non-negative value (the left side) is less than or equal to a negative value (the right side). Does that sound right to you?
Think about it this way: can a number that is zero or positive ever be less than a number that is negative? No, it can't! This is a contradiction. A non-negative number will always be greater than or equal to any negative number.
This realization is the key to solving this inequality. Because of the nature of square roots and the comparison we're making, we can already see that there might not be a solution. But let's go through the steps to formally prove this.
Step-by-Step Solution
Even though we suspect there's no solution, let's go through the typical steps you might take to solve an inequality like this. This will help solidify our understanding and show why the contradiction arises.
Step 1: Consider the domain
Before we even start manipulating the inequality, we need to think about the domain, which is the set of all possible values of 'x' that make the expression defined. In this case, we have a couple of square roots, so we need to make sure the expressions inside the square roots are non-negative.
First, we have √x²-16 inside the outer square root. This means x²-16 must be greater than or equal to zero: x² - 16 ≥ 0
We can factor this as a difference of squares: (x - 4)(x + 4) ≥ 0
To solve this inequality, we can use a sign chart or consider the intervals where the expression is positive or zero. The critical points are x = -4 and x = 4. Testing values in the intervals, we find that the inequality holds when: x ≤ -4 or x ≥ 4
So, the domain is all real numbers less than or equal to -4 or greater than or equal to 4. This is important because any solution we find must be within this domain.
Step 2: Analyze the Inequality
Now, let's go back to our original inequality: √√x²-16 ≤ -5
As we discussed, the left side of the inequality, √√x²-16, represents a non-negative value. The square root of any real number (in the principal sense) is always greater than or equal to zero.
The right side of the inequality is -5, which is a negative number.
Step 3: The Contradiction
Here's the crucial point: we're saying that a non-negative value (√√x²-16) is less than or equal to a negative value (-5). This is impossible!
A non-negative number can never be less than a negative number. Therefore, there is a contradiction, and there is no solution to this inequality.
Step 4: Formal Conclusion
We can formally state that the inequality √√x²-16 ≤ -5 has no solution. There is no value of 'x' that will satisfy this inequality.
Why No Solution?
Let's recap why there's no solution. The key lies in understanding the nature of square roots. The principal square root of a number is always non-negative. Because of this, the left side of our inequality, √√x²-16, will always be greater than or equal to zero.
On the other hand, the right side of the inequality is -5, which is a negative number. An inequality states that one value is less than or equal to another. But a non-negative value can never be less than a negative value. This fundamental contradiction means that there's no possible value of 'x' that can make the inequality true.
It's like saying you have a box of apples, and you're trying to figure out when the number of apples in the box is less than -5. Since you can't have a negative number of apples, it's just not possible!
Common Mistakes to Avoid
When dealing with inequalities involving square roots, there are a few common mistakes you should watch out for:
- Forgetting about the domain: It's crucial to consider the domain of the expression, especially when dealing with square roots. The expression inside the square root must be non-negative. Failing to consider the domain can lead to incorrect solutions.
- Assuming square roots can be negative: Remember, we're dealing with principal square roots, which are always non-negative. Don't make the mistake of thinking that √9 can be -3. It's only +3.
- Not recognizing contradictions: In this case, the key to solving the problem was recognizing the contradiction between a non-negative value and a negative value. Always be on the lookout for such contradictions, as they can often simplify the problem significantly.
- Squaring both sides without considering signs: If you were to try to solve this inequality by squaring both sides (which you shouldn't in this case), you'd need to be very careful about the signs. Squaring both sides can introduce extraneous solutions if you don't account for the fact that squaring can make a negative number positive.
Conclusion
So, there you have it! The inequality √√x²-16 ≤ -5 has no solution. This is because the left side of the inequality is always non-negative, while the right side is negative, creating a contradiction. Understanding the fundamental properties of square roots and being mindful of the domain are crucial for solving problems like this.
I hope this step-by-step guide has been helpful! Remember, math can be fun, especially when you break it down into smaller, more manageable pieces. Keep practicing, and you'll become a pro at solving inequalities in no time! If you have any questions or want to try another problem, let me know in the comments below. Happy math-ing, guys!
