Finding Integer Solutions To 1 ≤ X² ≤ 9 A Comprehensive Guide

Hey guys! Today, we're diving into a fun little math problem where we need to figure out all the integer values of x that make the inequality 1 ≤ x² ≤ 9 true. It might sound a bit intimidating at first, but trust me, it's totally manageable and actually pretty interesting once you break it down. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we're all on the same page. The key here is the inequality 1 ≤ x² ≤ 9. What this means is that we're looking for numbers x such that when you square them (multiply them by themselves), the result is greater than or equal to 1 and less than or equal to 9. The other important piece of information is that our domain is Z, which represents the set of integers. Integers are all the whole numbers (no fractions or decimals) and their negatives, including zero. So, we're talking about numbers like ..., -3, -2, -1, 0, 1, 2, 3, ...

The biggest challenge here might be that we are dealing with x² rather than just x. Squaring a number can change things up a bit, especially when we're considering negative numbers. Remember, a negative number squared becomes positive (e.g., (-2)² = 4). This means we need to think about both positive and negative values of x when finding our solutions. This also implies that there can be two different values of x who have the same result when squared, such as 2 and -2.

So, to recap, our mission is to find all the integers x that, when squared, fall within the range of 1 to 9, inclusive. We'll need to consider both positive and negative integers to make sure we've got the complete set of solutions. Let's dive into how we can actually do that!

Breaking Down the Inequality

To solve the compound inequality 1 ≤ x² ≤ 9, let’s break it down into two simpler inequalities. This will make it easier to manage and visualize the possible values of x. We can think of the original inequality as a combination of two separate conditions:

1. 1 ≤ x²: This part tells us that the square of x must be greater than or equal to 1.
2. x² ≤ 9: This part tells us that the square of x must be less than or equal to 9.

Now, let's tackle each of these individually. For the first inequality, 1 ≤ x², we need to find all integers whose square is 1 or greater. Think about it: If x is 0, then x² is 0, which is not greater than or equal to 1. So, 0 is out. But if x is 1, then x² is 1, which is greater than or equal to 1. So, 1 is a possibility. Similarly, if x is -1, then x² is also 1, which fits the bill.

As the absolute value of x increases (i.e., as x moves further away from 0 in either the positive or negative direction), x² will also increase. This means that any integer with an absolute value of 1 or more will satisfy the first inequality. For the second inequality, x² ≤ 9, we're looking for integers whose square is 9 or less. This puts an upper limit on the values of x we can consider. For example, if x is 4, then x² is 16, which is greater than 9. So, 4 (and any integer with a larger absolute value) won't work. But if x is 3, then x² is 9, which is right on the mark. Similarly, if x is -3, then x² is also 9.

By breaking the original inequality into these two parts, we've created a clearer picture of the constraints on x. We know that x² must be at least 1, and it can't be more than 9. Now, we just need to find the specific integers that fit both criteria.

Finding the Integer Solutions

Now comes the fun part: figuring out which integers actually satisfy both 1 ≤ x² and x² ≤ 9. We've already laid the groundwork by breaking down the inequality into two simpler conditions. Let's systematically consider integers and see if their squares fall within the range of 1 to 9.

Let's start with 0. We already know that 0 doesn't work because 0² is 0, which is less than 1. So, we can rule out 0 right away. Next, let's try 1. We have 1² = 1, which satisfies both 1 ≤ x² and x² ≤ 9. So, 1 is definitely in our solution set. Since we know that squaring a negative number gives the same result as squaring its positive counterpart, let's check -1. We have (-1)² = 1, which also satisfies both inequalities. So, -1 is in our solution set too.

Moving on to 2, we have 2² = 4. This is greater than or equal to 1 and less than or equal to 9, so 2 is a valid solution. Again, we know that (-2)² will also be 4, so -2 is also in our solution set. Let's try 3. We have 3² = 9, which satisfies both inequalities. This means 3 is a solution, and so is -3 because (-3)² = 9. Now, let's consider 4. We have 4² = 16, which is greater than 9. This violates the second inequality, x² ≤ 9, so 4 is not a solution. And since squaring any integer with an absolute value greater than 3 will result in a number greater than 9, we can stop here. We've essentially found all the integers that work.

So, the integers that satisfy both 1 ≤ x² and x² ≤ 9 are -3, -2, -1, 1, 2, and 3. These are the numbers whose squares fall within the range of 1 to 9.

Expressing the Truth Set

We've done the hard work of identifying all the integers that satisfy the inequality 1 ≤ x² ≤ 9. Now, let's express our solution in a formal way. In mathematics, the set of all solutions to a predicate (like an inequality) is called the truth set. We typically write a set using curly braces {} and list the elements (in this case, integers) separated by commas.

Based on our exploration in the previous section, we found that the integers -3, -2, -1, 1, 2, and 3 all satisfy the given inequality. Therefore, we can write the truth set as follows:

{-3, -2, -1, 1, 2, 3}

This is the complete set of integer solutions to the predicate 1 ≤ x² ≤ 9. It tells us exactly which integers, when squared, will fall within the specified range. Notice that we've included both the positive and negative counterparts for each number (except for 0, which wasn't a solution in the first place). This is crucial because squaring a negative integer results in a positive value, so we need to consider both possibilities.

So, to summarize, the truth set {-3, -2, -1, 1, 2, 3} is our final answer. It's a concise and precise way of expressing all the integer solutions to the given inequality. We've successfully found all the values of x that make the predicate true!

Conclusion

And there you have it, guys! We've successfully navigated through the inequality 1 ≤ x² ≤ 9 and found its truth set within the domain of integers. We started by understanding the problem, breaking it down into simpler parts, systematically testing integers, and finally expressing our solution as a set. Along the way, we reinforced the importance of considering both positive and negative values when dealing with squares and inequalities.

This type of problem is a great example of how mathematical concepts connect and build upon each other. We used our understanding of inequalities, squares, integers, and set notation to arrive at the solution. It's also a reminder that even seemingly complex problems can be tackled by breaking them down into smaller, more manageable steps.

So, the next time you encounter an inequality or any mathematical challenge, remember to take a deep breath, break it down, and approach it systematically. You might be surprised at how much you can accomplish! Keep practicing, keep exploring, and keep those mathematical gears turning. You got this!