Solving Quadratic Inequality X²-5≤0 A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of quadratic inequalities and explore how to solve them. Today, we're tackling the inequality x² - 5 ≤ 0. Our mission is to find the set of all 'x' values that make this inequality true. Think of it as a puzzle where we need to find the hidden range of numbers that satisfy the condition. We'll break down the steps, making it super easy to understand. So, buckle up and let's embark on this mathematical adventure together!

Understanding the Quadratic Inequality

At its heart, the quadratic inequality x² - 5 ≤ 0 is a comparison. We're looking for all the values of x that, when squared and then have 5 subtracted, result in a number that is either less than or equal to zero. To truly grasp this, let's visualize it. Imagine a number line stretching infinitely in both directions. Our goal is to pinpoint the section(s) of this line where the inequality holds true. This region, or these regions, will form our solution set. To begin, we need to transform the inequality into a more manageable form. The key is to find the critical points – the values of x where the expression x² - 5 equals zero. These points act as boundaries, dividing the number line into intervals that we can then test to see if they satisfy the original inequality. Think of it as setting up checkpoints along our number line journey. Once we know these critical points, we can systematically explore each interval to uncover the solution set. So, let's roll up our sleeves and find those critical points!

Finding the Critical Points

The critical points of the inequality are the values of x that make the expression x² - 5 equal to zero. To find them, we need to solve the quadratic equation x² - 5 = 0. This is where our algebraic skills come into play! One way to solve this equation is by isolating x² on one side. We can do this by adding 5 to both sides, giving us x² = 5. Now, to get x by itself, we need to take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will result in a positive number. So, we have x = ±√5. This means our critical points are √5 and -√5. These two values are crucial because they divide the number line into three distinct intervals: values less than -√5, values between -√5 and √5, and values greater than √5. Each of these intervals needs to be tested to see if it satisfies the original inequality. Think of √5 and -√5 as the gatekeepers, and we need to check which side of the gate the solutions lie on. Let's move on to the next step and test these intervals!

Testing the Intervals

Now that we've identified our critical points, √5 and -√5, the next step is to test the intervals they create on the number line. These intervals are: x < -√5, -√5 < x < √5, and x > √5. To test each interval, we'll pick a test value within the interval and plug it into our original inequality, x² - 5 ≤ 0. If the inequality holds true for the test value, then the entire interval is part of our solution set. Let's start with the interval x < -√5. A convenient test value here might be x = -3 (since -3 < -√5 ≈ -2.24). Plugging this into our inequality, we get (-3)² - 5 = 9 - 5 = 4. Since 4 is not less than or equal to 0, this interval is not part of our solution. Next, let's test the interval -√5 < x < √5. A simple test value here would be x = 0. Plugging this in, we get (0)² - 5 = -5. Since -5 is less than or equal to 0, this interval is part of our solution. Finally, let's test the interval x > √5. A suitable test value here is x = 3 (since 3 > √5 ≈ 2.24). Plugging this in, we get (3)² - 5 = 9 - 5 = 4. Again, 4 is not less than or equal to 0, so this interval is not part of our solution. But we're not quite done yet! We also need to consider the critical points themselves. Since our inequality includes "equal to", we need to check if x = -√5 and x = √5 are also solutions. Plugging these values into the inequality, we see that both satisfy the condition. This is super important, guys, because it means we need to include these endpoints in our final solution set. Now we're ready to put it all together and express our solution!

Expressing the Solution Set

We've done the legwork, and now it's time to express our solution set in a clear and concise way. Remember, our solution includes the interval -√5 < x < √5 and the critical points x = -√5 and x = √5. This means that all values of x between -√5 and √5, inclusive, satisfy the inequality x² - 5 ≤ 0. There are a couple of ways we can represent this mathematically. One common way is to use interval notation. In interval notation, we use brackets to indicate that the endpoints are included in the solution and parentheses to indicate that they are not. Since our solution includes both -√5 and √5, we'll use brackets. So, in interval notation, our solution set is written as [-√5, √5]. This notation neatly encapsulates the range of values that make our inequality true. Another way to represent the solution set is using inequality notation. In this notation, we use symbols like ≤ and ≥ to show the range of values. In our case, we can express the solution as -√5 ≤ x ≤ √5. This notation directly states that x is greater than or equal to -√5 and less than or equal to √5. Both interval and inequality notation are valuable tools for expressing solution sets, and the choice of which to use often comes down to personal preference or the specific context of the problem. Now that we've successfully navigated this inequality, let's take a moment to reflect on the general process and see how we can apply these skills to other problems.

Final Solution

Therefore, the solution set of the quadratic inequality x² - 5 ≤ 0 is O {x | -√5 ≤ x ≤ √5}. This means that any value of x within this range, including the endpoints -√5 and √5, will satisfy the original inequality. We've successfully navigated the world of quadratic inequalities and emerged victorious! But the adventure doesn't end here. The principles and techniques we've learned can be applied to a wide range of mathematical problems. The key is to break down complex problems into smaller, more manageable steps. Finding critical points, testing intervals, and expressing the solution set are all powerful tools in your mathematical arsenal. So, keep practicing, keep exploring, and keep pushing the boundaries of your mathematical understanding. Who knows what challenges you'll conquer next!

General Approach to Solving Quadratic Inequalities

Okay, guys, let's zoom out for a moment and talk about the general strategy for tackling quadratic inequalities. We've successfully solved x² - 5 ≤ 0, but the real power comes from understanding the process so we can apply it to other inequalities. The key steps are: 1. Rewrite the inequality: If necessary, rearrange the inequality so that one side is zero. This puts it in a standard form that's easier to work with. For example, if you have something like x² + 2x > 3, you'd subtract 3 from both sides to get x² + 2x - 3 > 0. 2. Find the critical points: These are the values of x that make the quadratic expression equal to zero. You find them by solving the quadratic equation. Remember, you can use factoring, the quadratic formula, or completing the square to solve the equation. The critical points are like the anchor points that define the intervals on the number line. 3. Create intervals: The critical points divide the number line into intervals. These intervals are the regions we need to test. 4. Test each interval: Pick a test value within each interval and plug it into the original inequality. If the inequality is true for the test value, then the entire interval is part of the solution set. 5. Consider the endpoints: If the inequality includes "equal to" (≤ or ≥), then the critical points themselves are also part of the solution. You need to include them in your final answer. 6. Express the solution: Write the solution set using interval notation or inequality notation. This clearly communicates the range of values that satisfy the inequality. By following these steps, you can confidently solve a wide variety of quadratic inequalities. Think of it as a recipe – follow the instructions, and you'll get the right result! And remember, practice makes perfect. The more you work with these concepts, the more comfortable you'll become.

Conclusion

So there you have it, guys! We've successfully unraveled the mystery of the quadratic inequality x² - 5 ≤ 0. We've journeyed from understanding the inequality itself to finding critical points, testing intervals, and expressing the final solution set. We even took a step back to discuss the general approach to solving quadratic inequalities, arming ourselves with a versatile strategy for future challenges. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing the skills to tackle new and exciting problems. The solution set O {x | -√5 ≤ x ≤ √5} represents a range of values that satisfy our inequality, but it also symbolizes our journey through the world of quadratic inequalities. Keep exploring, keep questioning, and keep pushing the boundaries of your mathematical knowledge. The world of mathematics is vast and full of wonders waiting to be discovered!