Solving Double Inequalities: Find The Solution Set

Hey guys! Let's dive into the world of double inequalities! If you've ever stumbled upon an inequality that looks like a sandwich – with a variable trapped between two bounds – you're in the right place. We're going to break down how to solve these problems step-by-step, making it super easy to understand. Double inequalities might seem intimidating at first, but trust me, they're totally manageable once you grasp the basic concepts. We'll walk through the process, from understanding the notation to finding the solution set and even visualizing it on a number line. Get ready to boost your algebra skills!

Understanding Double Inequalities

In essence, double inequalities are a compact way of writing two inequalities at once. Think of it like this: instead of writing -2 < x - 7 and x - 7 ≤ 5 separately, we combine them into a single expression: -2 < x - 7 ≤ 5. The key here is that the variable, in this case x, must satisfy both inequalities simultaneously. It's like having two conditions to meet at the same time. The notation is crucial; it tells us the relationship between the three parts of the inequality. The "less than" symbol (<) means x - 7 is greater than -2, but not equal to it. The "less than or equal to" symbol (≤) means x - 7 is less than or equal to 5, allowing for the possibility of equality. This combined statement gives us a range within which x must fall. A solid understanding of this notation is the foundation for solving any double inequality. Without it, we'd be trying to navigate a maze blindfolded! So, let's make sure we're crystal clear on what these symbols mean and how they connect to form a single, powerful mathematical statement. Mastering this will make the rest of the process much smoother, and you'll be solving these problems like a pro in no time!

Breaking Down the Notation

Let's really nail down the notation used in double inequalities. This is super important because understanding the symbols is the first step to cracking these problems. The inequality -2 < x - 7 ≤ 5 is actually two inequalities mashed together. The first part, -2 < x - 7, tells us that x - 7 is greater than -2. Remember, the "<" symbol means "less than," but when we read it from right to left, it becomes "greater than." So, think of it as -2 being smaller than the expression x - 7. Now, the second part, x - 7 ≤ 5, says that x - 7 is less than or equal to 5. The "≤" symbol includes the possibility of equality, meaning x - 7 can be 5. The magic of a double inequality is that it combines these two conditions. The value of x - 7 has to be both greater than -2 and less than or equal to 5. It's like saying, "You have to be taller than this line but shorter than that line." This "betweenness" is what defines double inequalities. Getting comfortable with this notation is like learning the alphabet of algebra. Once you know the letters, you can start forming words, and in this case, solving inequalities! So, take a moment to let this sink in, and you'll be well on your way to mastering these problems. We're building a solid foundation here, brick by brick, and this notational understanding is a key cornerstone.

Why Double Inequalities Matter

Why should you even care about double inequalities? Well, they pop up in all sorts of real-world situations and higher-level math problems. Think about setting limits – maybe you want a temperature to stay within a certain range, or you need a budget to fall between two amounts. Double inequalities are the perfect tool for expressing these kinds of constraints. They're not just abstract math concepts; they're a way to describe and solve problems that involve ranges and boundaries. In calculus, you'll see them used to define intervals and limits, which are fundamental to understanding derivatives and integrals. In statistics, they help define confidence intervals, which tell us the range within which a population parameter is likely to fall. Even in everyday life, you might use a double inequality without realizing it. For example, if you say you want to arrive at a meeting between 9:00 AM and 9:15 AM, you're essentially setting a double inequality for your arrival time. So, learning to solve these inequalities isn't just about passing a test; it's about gaining a powerful tool for problem-solving in a variety of contexts. It's like learning a new language – once you're fluent, you can communicate in all sorts of new situations. The more you understand double inequalities, the more equipped you'll be to tackle complex problems and see the world in a more mathematical way. This is about building a skill that will serve you well beyond the classroom.

Solving the Inequality Step-by-Step

Alright, let's get down to the nitty-gritty and solve the inequality -2 < x - 7 ≤ 5. The goal here is to isolate x in the middle, just like we would when solving a regular equation. The trick is that whatever we do to one part of the inequality, we have to do to all parts. It's like a balancing act – we need to keep the inequality "in balance" to maintain its truth. The first step is to get rid of that pesky -7 next to the x. To do that, we'll add 7 to all three parts of the inequality. This is the golden rule of solving inequalities: perform the same operation on all sides. Adding 7 to -2 gives us 5, adding 7 to x - 7 gives us x, and adding 7 to 5 gives us 12. So, our inequality now looks like this: 5 < x ≤ 12. Ta-da! We've isolated x. This tells us that x is greater than 5 but less than or equal to 12. That's our solution set! We've successfully navigated the algebraic terrain and found the range of values that satisfy the original double inequality. This step-by-step approach is the key to demystifying these problems. By breaking it down into smaller, manageable steps, we can conquer even the most intimidating inequalities. Now, let's explore how to interpret this solution and represent it visually.

Isolating the Variable

Isolating the variable is the name of the game when solving any inequality, including double inequalities. Think of it like solving a puzzle – we want to get x all by itself in the middle so we can clearly see its possible values. The key is to perform inverse operations, just like we do with regular equations, but we have to apply them to all three parts of the inequality. In our example, -2 < x - 7 ≤ 5, we had that pesky -7 hanging out with the x. To get rid of it, we used the inverse operation: adding 7. We added 7 to the left side (-2), the middle (x - 7), and the right side (5). This is crucial! If we only added 7 to two parts, we'd be changing the entire relationship and messing up the solution. It's like trying to balance a scale by adding weight to only one side – it just won't work. By adding 7 to all parts, we maintained the balance and kept the inequality true. This resulted in 5 < x ≤ 12, where x is now isolated. We can clearly see that x must be greater than 5 and less than or equal to 12. This process of isolation is the heart of solving inequalities. It's about strategically using inverse operations to peel away the layers surrounding the variable until it stands alone, revealing its solution. Master this technique, and you'll be well-equipped to tackle any inequality that comes your way.

Interpreting the Solution Set

Once we've isolated x, we need to interpret the solution set. This means understanding what the inequality 5 < x ≤ 12 actually tells us. It's not just a string of symbols; it's a statement about the possible values of x. The solution 5 < x means that x can be any number greater than 5. It can be 5.0001, 5.1, 6, 10, or a million – as long as it's bigger than 5. However, it cannot be 5 itself. This is crucial because the "<" symbol means strictly greater than. Now, the solution x ≤ 12 means that x can be any number less than or equal to 12. It can be 12, 11.99, 10, 5, or even -100. The "≤" symbol includes the possibility of equality, so 12 is a valid solution. Combining these two conditions, 5 < x ≤ 12, we get a range of values. x must be bigger than 5 and less than or equal to 12. Think of it as a restricted zone – x can hang out anywhere between 5 and 12, including 12, but not including 5. To really solidify this understanding, it's helpful to visualize the solution on a number line, which we'll discuss next. But for now, make sure you're comfortable with interpreting the inequality symbols and understanding the range of values they define. This is the key to unlocking the meaning behind the mathematical expression.

Visualizing the Solution on a Number Line

Visualizing the solution on a number line is a fantastic way to make the concept of double inequalities even clearer. A number line is simply a line that represents all real numbers, with zero in the middle, positive numbers to the right, and negative numbers to the left. We can use this line to graphically represent the solution set of our inequality, 5 < x ≤ 12. First, we locate the key numbers, 5 and 12, on the number line. Now, we need to represent the inequalities themselves. Since x is greater than 5, we use an open circle at 5. This open circle indicates that 5 is not included in the solution set. It's like a boundary that x can't cross. For the other end, x is less than or equal to 12, so we use a closed circle at 12. This closed circle means that 12 is included in the solution set. It's a boundary that x can touch. Finally, we shade the region between the open circle at 5 and the closed circle at 12. This shaded region represents all the numbers that satisfy the double inequality. Any number within this shaded area is a valid solution. The number line provides a visual representation of the range of values that x can take. It's a powerful tool for understanding and communicating the solution set of an inequality. By seeing the solution graphically, we can often grasp the concept more intuitively than by simply looking at the algebraic expression. So, grab a piece of paper and try visualizing other double inequalities on a number line – you'll be amazed at how much it helps!

Open vs. Closed Circles

Let's zoom in on those circles we use on the number line – the open circles and closed circles. These little guys are super important because they tell us whether or not the endpoint of an interval is included in the solution set. Remember, an open circle means the endpoint is not included, while a closed circle means it is included. This distinction comes directly from the inequality symbols themselves. If we have a "<" (less than) or a ">" (greater than) symbol, we use an open circle. These symbols mean the variable can get infinitely close to the endpoint, but it can't actually be equal to it. It's like an invisible barrier. On the other hand, if we have a "≤" (less than or equal to) or a "≥" (greater than or equal to) symbol, we use a closed circle. These symbols include the possibility of equality, so the endpoint is a valid solution. It's a solid point that the variable can land on. In our example, 5 < x ≤ 12, we used an open circle at 5 because x is strictly greater than 5. We used a closed circle at 12 because x can be equal to 12. Getting these circles right is crucial for accurately representing the solution set on a number line. A misplaced circle can completely change the meaning of the graph. So, always double-check your inequality symbols and make sure your circles match! It's a small detail that makes a big difference in understanding and communicating the solution.

Shading the Solution Region

Once you've placed your open and closed circles on the number line, the next step is to shade the solution region. This is where you visually represent all the numbers that satisfy the inequality. The shaded region is the area between the circles, and it represents the range of possible values for the variable. To determine which region to shade, think about what the inequality is telling you. In our example, 5 < x ≤ 12, x has to be greater than 5 and less than or equal to 12. So, we need to shade the area between 5 and 12. This shaded area includes all the numbers between 5 and 12, excluding 5 (due to the open circle) and including 12 (due to the closed circle). It's like painting a picture of the solution set. The shaded region makes it immediately clear which numbers are valid solutions and which are not. If you were to pick any number within the shaded region, it would satisfy the original inequality. This visual representation is incredibly helpful for understanding the concept of a solution set and for communicating it to others. It's a quick and easy way to convey a lot of information. So, grab your metaphorical paintbrush and shade those number lines – you're creating a visual masterpiece of mathematical understanding!

Applying the Solution to the Given Options

Now that we've solved the inequality 5 < x ≤ 12, let's apply our solution to the given options: a) 5; b) 10; c) 1; d) 9. Remember, the solution set includes all numbers greater than 5 and less than or equal to 12. So, we need to check each option to see if it falls within this range. Option a) is 5. Is 5 greater than 5? No. Therefore, 5 is not a solution. Option b) is 10. Is 10 greater than 5? Yes. Is 10 less than or equal to 12? Yes. Therefore, 10 is a solution. Option c) is 1. Is 1 greater than 5? No. Therefore, 1 is not a solution. Option d) is 9. Is 9 greater than 5? Yes. Is 9 less than or equal to 12? Yes. Therefore, 9 is a solution. So, the numbers that are solutions to the double inequality are 10 and 9. We've successfully used our understanding of the solution set to identify the correct answers from the given options. This is a crucial skill in algebra – being able to translate a mathematical solution into a concrete answer. It's like connecting the dots between the abstract world of equations and the real world of numbers. Now, let's solidify this understanding with a quick recap.

Checking Each Option

Checking each option against our solution set is like being a detective, matching clues to a suspect. We have our solution, 5 < x ≤ 12, and we have a lineup of potential solutions: 5, 10, 1, and 9. Our job is to see which ones fit the criteria. The solution 5 < x ≤ 12 has two parts: x must be greater than 5, and x must be less than or equal to 12. We need to check both conditions for each option. Let's start with 5. Is 5 greater than 5? No. So, 5 fails the first test and is eliminated. Next, let's look at 10. Is 10 greater than 5? Yes. Is 10 less than or equal to 12? Yes. So, 10 passes both tests and is a solution! Now, let's consider 1. Is 1 greater than 5? No. So, 1 fails the first test and is out. Finally, let's check 9. Is 9 greater than 5? Yes. Is 9 less than or equal to 12? Yes. So, 9 passes both tests and is also a solution! By systematically checking each option against our solution set, we can confidently identify the correct answers. This process of verification is a crucial step in problem-solving. It's like proofreading your work to catch any errors. Always take the time to check your answers – it's a sure way to boost your confidence and accuracy.

Identifying the Correct Answers

After meticulously identifying the correct answers by checking each option, we can now confidently state the solutions to our problem. We found that 10 and 9 are the numbers that satisfy the double inequality -2 < x - 7 ≤ 5. These numbers fall within the range defined by our solution set, 5 < x ≤ 12. They are greater than 5 and less than or equal to 12. This final step is the culmination of our problem-solving journey. We started by understanding the concept of double inequalities, then we learned how to solve them algebraically, visualize the solution on a number line, and finally, apply our solution to specific options. By following this step-by-step process, we've not only found the correct answers but also deepened our understanding of the underlying mathematical principles. This is what true problem-solving is all about – not just getting the right answer, but understanding why it's the right answer. So, congratulations on making it to the finish line! You've successfully navigated the world of double inequalities and emerged victorious. Now, go forth and conquer other mathematical challenges with confidence and skill!

Conclusion

So there you have it, guys! We've tackled double inequalities head-on and emerged victorious! We started by understanding what these inequalities mean, then we learned how to solve them step-by-step, visualize the solution on a number line, and finally, apply our knowledge to find the correct answers. Remember, the key to success with double inequalities is to break them down into manageable steps. Isolate the variable by performing the same operations on all three parts of the inequality. Interpret the solution set carefully, paying attention to the inequality symbols. Visualize the solution on a number line to gain a deeper understanding. And finally, always check your answers to ensure accuracy. By mastering these techniques, you'll be well-equipped to tackle any double inequality that comes your way. Math can be challenging, but with a systematic approach and a little bit of practice, you can conquer any problem. Keep practicing, keep learning, and keep pushing yourself to grow. You've got this! Now, go out there and show those inequalities who's boss! This journey through double inequalities is a testament to your problem-solving abilities. You've learned a valuable skill that will serve you well in future math endeavors. So, celebrate your success and get ready for the next challenge!