Solving Inequalities: A Comprehensive Guide

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Hey guys! Let's dive into the world of inequalities and how to solve them. Understanding inequalities is super important, especially if you're tackling math problems or just trying to figure out real-world scenarios. We'll break down the problem you've got, "√(x² + 5) * (x + 4) ≥ 0", step by step. So, grab a coffee (or your favorite drink), and let's get started!

Understanding the Basics of Inequalities

Alright, before we get our hands dirty with the actual problem, let's make sure we're all on the same page about what inequalities are. Basically, they're mathematical statements that show a relationship between two expressions that aren't equal. Instead of an equals sign (=), we use symbols like:

  • > (greater than)
  • < (less than)
  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)

Now, the main goal when solving an inequality is to find the range of values that make the statement true. This range is usually expressed as an interval, which is what those answer choices (A, B, C, D) in your problem are all about. Think of it like this: you're trying to figure out which numbers, when plugged into the inequality, will make it a true statement. It's like a treasure hunt, and the solution is the treasure!

One important thing to remember is that solving inequalities is very similar to solving equations, but there's a slight twist. When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have x > 2 and you multiply both sides by -1, you get -x < -2. Got it? Cool!

Now let's break down the given problem and then look at the answer choices. Keep in mind, the key to solving any inequality is to isolate the variable (in this case, 'x') and figure out the range of values that satisfy the condition. The more you practice, the easier this process will become. Don't worry if it seems tricky at first; we'll break it down into manageable chunks.

Now, let's look closely at our problem: "√(x² + 5) * (x + 4) ≥ 0". This is a product of two factors, and we need to figure out when that product is greater than or equal to zero. This happens when both factors are positive or when both factors are negative.

Decoding the Inequality: √(x² + 5) * (x + 4) ≥ 0

Okay, so we're looking at the inequality: √(x² + 5) * (x + 4) ≥ 0. It looks a little intimidating at first glance, but let's break it down into smaller, more manageable pieces, alright? This approach makes the whole process less scary and easier to handle.

First, consider the term √(x² + 5). This is a square root. Here's a neat trick to keep in mind: the expression inside the square root, which is x² + 5, will always be positive. Why? Because x² is always non-negative (it's either zero or positive), and we're adding 5 to it, so the sum will always be at least 5. This means that the square root is always a real number and, in fact, always positive or zero. This is a huge simplification! It means we don't need to worry about the square root term causing any issues with the inequality's sign.

So, we can essentially ignore the square root part when determining the overall sign of the product, because it's always positive. Our focus should be on the other factor: (x + 4). For the entire expression √(x² + 5) * (x + 4) to be greater than or equal to zero, (x + 4) must also be greater than or equal to zero. Remember, the product of two positive numbers (or zero times anything) is always positive (or zero). And since our square root term is always positive, the sign of the entire expression depends only on the sign of the (x + 4) term.

Therefore, we need to solve the inequality x + 4 ≥ 0. This is way easier than the original problem. To solve it, we simply isolate x:

Subtract 4 from both sides: x ≥ -4

This tells us that x must be greater than or equal to -4 for the original inequality to hold true. The solution is all the numbers from -4 to positive infinity. Now, let’s go back to those multiple-choice answers and see which one matches our solution.

Finding the Solution Set

Alright, we've done the hard work, now let's find the solution! Remember that after our simplification, we discovered that x ≥ -4. That's the set of all real numbers greater than or equal to -4. Now, let’s see which of the provided answer options matches this:

  • A. (-∞; -4]: This means all numbers from negative infinity up to and including -4. This is not what we want because we need numbers greater than or equal to -4.
  • B. [-4; +∞): This means all numbers from -4 (inclusive) to positive infinity. This is exactly what we want, because it includes all the numbers that are greater than or equal to -4.
  • C. [4; +∞): This means all numbers from 4 (inclusive) to positive infinity. This is incorrect, as our solution includes numbers starting from -4.
  • D. (-∞; 4]: This means all numbers from negative infinity up to and including 4. This is incorrect, as it includes numbers less than -4 which don't satisfy our original inequality.

So, the correct answer is B. [-4; +∞). This interval notation accurately represents all the x-values that make the inequality true. The square bracket [ means that -4 is included in the solution set, while the parenthesis ) means that positive infinity is not a specific number, but the set continues endlessly. Congratulations, you found the solution!

Conclusion: Putting It All Together

Awesome work, guys! We've successfully navigated through the inequality problem: "√(x² + 5) * (x + 4) ≥ 0". Here’s a quick recap of what we did:

  1. Analyzed the problem: We identified that the square root part was always positive, simplifying the problem.
  2. Isolated the critical factor: We realized that the sign of the entire expression depended on the (x + 4) term.
  3. Solved for x: We found that x ≥ -4.
  4. Matched with the answer choices: We saw that the correct answer was B. [-4; +∞), representing all numbers from -4 to positive infinity.

Inequalities can seem daunting at first, but remember to break them down into smaller, manageable parts. Understand the basic rules, and you'll be well on your way to mastering these kinds of problems. Keep practicing, and you'll become a pro in no time! Keep up the great work, and don't be afraid to ask for help or clarification. Math is a journey, and we're all in it together!