Solving Inequalities And Intervals: Finding The Range Of (x+1)/(x+2)
Introduction: Unveiling the Realm of Inequalities
Hey guys! Today, we're diving deep into the fascinating world of inequalities and interval notation. We're going to tackle a problem that might seem a bit daunting at first, but I promise, we'll break it down step-by-step until it becomes crystal clear. Our main goal is to find the interval of the expression (x+1)/(x+2) given that 2x+5/-3 falls within the interval [5,8). This involves manipulating inequalities, solving for x, and understanding how to express our solution in interval notation. So, buckle up, and let's get started!
Before we jump into the specifics, let's quickly recap what inequalities and interval notation are all about. Inequalities are mathematical statements that compare two values, indicating that one is greater than, less than, or not equal to the other. We use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to represent these relationships. For example, the inequality x > 3 means that x can be any number greater than 3. Interval notation, on the other hand, is a concise way to represent a set of numbers within a specific range. We use brackets and parentheses to indicate whether the endpoints of the interval are included or excluded. A square bracket [ ] indicates that the endpoint is included, while a parenthesis ( ) indicates that it's excluded. For instance, the interval [2, 5) represents all numbers between 2 and 5, including 2 but excluding 5. Understanding these basics is crucial for tackling our main problem.
The problem we're going to solve combines these concepts beautifully. We have a compound inequality involving a rational expression, and we need to find the range of values for x that satisfy it. This is a classic example of a problem that tests our understanding of algebraic manipulation, inequality properties, and interval notation. By working through it, we'll not only sharpen our math skills but also gain a deeper appreciation for how these concepts connect in the real world. Think of it like this: inequalities are the building blocks, algebraic manipulation is the construction crew, and interval notation is the blueprint that helps us visualize the final structure. So, let's put on our hard hats and start building!
Step-by-Step Solution: Cracking the Inequality
Okay, let's get down to business! Our first task is to decipher the given information. We know that the expression (2x + 5) / -3 lies within the interval [5, 8). This means that the value of (2x + 5) / -3 is greater than or equal to 5 and less than 8. We can write this as a compound inequality: 5 ≤ (2x + 5) / -3 < 8. Now, our mission is to isolate x and find the range of values that satisfy this inequality. The key here is to remember the golden rule of inequalities: when you multiply or divide both sides by a negative number, you must flip the inequality signs. This is super important, so keep it in the back of your mind as we proceed.
To start, let's get rid of the fraction. We can do this by multiplying all parts of the inequality by -3. Remember the golden rule? Since we're multiplying by a negative number, we need to flip the inequality signs. So, we get: 5 * (-3) ≥ (2x + 5) / -3 * (-3) > 8 * (-3). This simplifies to -15 ≥ 2x + 5 > -24. Great! We've eliminated the fraction and now we have a more manageable inequality. Next, we want to isolate the term with x, which is 2x. To do this, we'll subtract 5 from all parts of the inequality: -15 - 5 ≥ 2x + 5 - 5 > -24 - 5. This simplifies to -20 ≥ 2x > -29. We're getting closer! Now, we just need to get x by itself. To do this, we'll divide all parts of the inequality by 2: -20 / 2 ≥ 2x / 2 > -29 / 2. This gives us -10 ≥ x > -14.5.
Awesome! We've successfully isolated x. But, before we celebrate, let's rewrite this inequality in a more conventional way. It's generally easier to read inequalities with the smaller value on the left. So, we can rewrite -10 ≥ x > -14.5 as -14.5 < x ≤ -10. This inequality tells us that x is greater than -14.5 and less than or equal to -10. Now, we need to express this solution in interval notation. Remember, parentheses indicate that the endpoint is excluded, and square brackets indicate that it's included. Since x is greater than -14.5, we'll use a parenthesis for that endpoint. And since x is less than or equal to -10, we'll use a square bracket for that endpoint. Therefore, the solution in interval notation is (-14.5, -10]. This interval represents all the possible values of x that satisfy the original inequality. We've cracked the code!
Diving Deeper: Analyzing (x+1)/(x+2)
Now that we've found the interval for x, which is (-14.5, -10], the next exciting challenge is to determine the corresponding interval for the expression (x + 1) / (x + 2). This is where things get a little more interesting because we're dealing with a rational function. Remember, a rational function is simply a fraction where both the numerator and the denominator are polynomials. To find the interval for this expression, we'll need to consider how the function behaves within our interval for x and also pay close attention to any potential points of discontinuity. A point of discontinuity occurs when the denominator of the fraction is equal to zero, as division by zero is undefined. So, let's keep that in mind as we proceed.
First, let's identify any potential points of discontinuity. The denominator of our expression is (x + 2). So, we need to find the value of x that makes x + 2 equal to zero. Solving the equation x + 2 = 0, we get x = -2. This means that the function (x + 1) / (x + 2) is undefined when x is -2. Now, we need to check if this point of discontinuity falls within our interval for x, which is (-14.5, -10]. Since -2 is indeed within this interval, we'll need to be extra careful when analyzing the function's behavior around this point. The presence of a discontinuity means that the function's value can change dramatically as x approaches -2, potentially affecting the resulting interval for the expression.
To determine the interval for (x + 1) / (x + 2), we can analyze the function's behavior at the endpoints of our interval for x and also consider the effect of the discontinuity at x = -2. Let's start by evaluating the expression at the endpoints. When x = -14.5, the expression becomes (-14.5 + 1) / (-14.5 + 2) = -13.5 / -12.5 = 1.08. When x = -10, the expression becomes (-10 + 1) / (-10 + 2) = -9 / -8 = 1.125. So, at the endpoints, the expression's value is around 1. Now, let's think about what happens near the discontinuity at x = -2. As x approaches -2 from the left (i.e., from values smaller than -2), the denominator (x + 2) becomes a very small negative number, and the numerator (x + 1) is close to -1. This means the expression becomes a large positive number. As x approaches -2 from the right (i.e., from values larger than -2), the denominator (x + 2) becomes a very small positive number, and the numerator (x + 1) is still close to -1. This means the expression becomes a large negative number. This behavior near the discontinuity tells us that the function's value can take on a wide range of values within our interval for x. Taking all of this into consideration, it becomes clear that we must analyze this further to accurately determine the interval. Analyzing the behavior of this function requires a more sophisticated approach, such as considering test points within the interval and analyzing the sign of the expression. Due to the complexity introduced by the discontinuity, accurately determining the range of (x+1)/(x+2) over the interval (-14.5, -10] requires a detailed analysis that goes beyond simply evaluating the endpoints. For a complete solution, techniques from calculus or more advanced algebraic methods might be necessary to fully understand the function's behavior and its range.
Conclusion: Mastering Inequalities and Intervals
Wow, guys! We've journeyed through a complex problem involving inequalities, rational expressions, and interval notation. We started by unraveling the given inequality, 5 ≤ (2x + 5) / -3 < 8, and skillfully isolated x to find its interval, which turned out to be (-14.5, -10]. This involved remembering the crucial rule of flipping inequality signs when multiplying or dividing by a negative number. Then, we took on the challenge of finding the interval for the expression (x + 1) / (x + 2), which introduced the concept of discontinuities and the need for careful analysis of the function's behavior. While we've laid the groundwork for this part of the problem, it's clear that a full solution requires more advanced techniques due to the discontinuity within our interval.
This exploration highlights the importance of a solid understanding of inequalities, interval notation, and the behavior of rational functions. These concepts are fundamental in mathematics and have wide-ranging applications in fields like physics, engineering, economics, and computer science. By tackling challenging problems like this one, we not only sharpen our problem-solving skills but also develop a deeper appreciation for the interconnectedness of mathematical ideas. Remember, guys, math isn't just about memorizing formulas; it's about understanding the underlying principles and using them to solve real-world problems. So, keep practicing, keep exploring, and never be afraid to dive into a challenging problem. The more you practice, the more confident you'll become in your ability to tackle any mathematical hurdle that comes your way!
