Solving The Inequality: (x+4)/(2x-1) < 0

Hey guys! Let's dive into how to solve the inequality: (x+4) / (2x-1) < 0. This is a classic problem in algebra, and understanding how to solve it is super important. We're essentially trying to find all the values of 'x' that make this fraction negative. It's a bit like a treasure hunt, but instead of gold, we're looking for the values of 'x' that satisfy this condition. The approach we'll use involves finding critical points, testing intervals, and understanding how the numerator and denominator affect the sign of the overall fraction. This method will help us in similar problems, so pay attention!

To begin, let's understand what makes a fraction negative. A fraction is negative if either the numerator (the top part) is positive and the denominator (the bottom part) is negative, or if the numerator is negative and the denominator is positive. This is the core principle that we'll use to solve this inequality. We need to find the values of 'x' where one of these two scenarios is true. Now, let’s get into the specifics. The first step involves identifying the critical points. These are the values of 'x' that make either the numerator or the denominator equal to zero. These points are really important because they divide the number line into intervals, and within each interval, the sign of the fraction remains constant. Let's break this down further.

First, we find when the numerator is zero. So, we solve the equation x + 4 = 0. Simple enough, right? Subtracting 4 from both sides, we get x = -4. This means that at x = -4, the numerator is zero. Next, we find when the denominator is zero. So, we solve the equation 2x - 1 = 0. Adding 1 to both sides gives us 2x = 1. Dividing both sides by 2, we get x = 1/2. At x = 1/2, the denominator is zero. Important note: The denominator cannot be zero because division by zero is undefined. This means that x = 1/2 is not part of our solution; it's a point where the function is undefined, but it's still crucial in our analysis because it’s a critical point that divides the number line into intervals. We will see that this is an important part of solving inequalities like this, it is really the only way to get a good result. Now that we have our critical points, x = -4 and x = 1/2, we can proceed to the next step. So, what’s next, you ask? We'll create intervals using these critical points and test the sign of the fraction within each interval. Are you excited to see how it works?

Finding Critical Points and Creating Intervals

Alright, let's get our hands dirty and determine the critical points and intervals. As we established earlier, the critical points are the values of 'x' that make either the numerator or the denominator equal to zero. This is where the fraction changes its behavior, so these points are super important. In our inequality, (x+4) / (2x-1) < 0, the critical points are where x + 4 = 0 and 2x - 1 = 0. We've already calculated these: x = -4 and x = 1/2. Remember, at x = -4, the numerator is zero, and at x = 1/2, the denominator is zero. The latter is especially important because it's a point where the function is undefined. Now that we have our critical points, we can create the intervals on the number line. These intervals are determined by the critical points, and they are the areas we'll test to see where the inequality holds true. These intervals are: (-∞, -4), (-4, 1/2), and (1/2, ∞). Note that -4 and 1/2 are the boundary points for the intervals. We'll test each of these intervals to find out where the inequality (x+4) / (2x-1) < 0 is true. Think of it like a map; we need to explore each region to find the treasure. Using intervals is a really useful method, so be sure to understand it well.

Let’s start with the first interval: (-∞, -4). This means we'll consider all values of 'x' less than -4. Let's pick a test value within this interval, say x = -5. We substitute x = -5 into the inequality (x+4) / (2x-1) < 0, to get (-5+4) / (2*-5-1) < 0. Simplifying, we get (-1) / (-11) < 0. This simplifies to 1/11 < 0, which is false. Therefore, the inequality is not true for any 'x' in the interval (-∞, -4). Now let's move on to the second interval: (-4, 1/2). This interval includes all values of 'x' between -4 and 1/2. Let's pick a test value here, say x = 0. Substituting x = 0 into the inequality, we get (0+4) / (20-1) < 0, which simplifies to 4 / (-1) < 0, or -4 < 0. This is true! This means that the inequality is true for all 'x' in the interval (-4, 1/2). Finally, let’s look at the third interval: (1/2, ∞). This interval includes all values of 'x' greater than 1/2. Let's pick a test value, say x = 1. Substituting x = 1 into the inequality, we get (1+4) / (21-1) < 0, which simplifies to 5/1 < 0, or 5 < 0. This is false. Therefore, the inequality is not true for any 'x' in the interval (1/2, ∞). Are you seeing how we're breaking down the problem step by step? We're almost there! In the next section, we'll summarize our results and state the solution.

Testing Intervals and Determining the Solution

Okay, guys, it's time to test those intervals and get to the solution. We have our intervals: (-∞, -4), (-4, 1/2), and (1/2, ∞). We've already picked some test values and substituted them into the original inequality (x+4) / (2x-1) < 0. Let's summarize our findings:

For the interval (-∞, -4), we tested x = -5. The result was (-1)/(-11) < 0, which simplifies to 1/11 < 0. This is false, meaning the inequality is not satisfied in this interval.

For the interval (-4, 1/2), we tested x = 0. The result was 4/(-1) < 0, which simplifies to -4 < 0. This is true! The inequality is satisfied in this interval.

For the interval (1/2, ∞), we tested x = 1. The result was 5/1 < 0, which simplifies to 5 < 0. This is false, meaning the inequality is not satisfied in this interval.

So, what does all this mean? It means that the inequality (x+4) / (2x-1) < 0 is only true in the interval (-4, 1/2). The solution to the inequality is therefore all values of 'x' that lie between -4 and 1/2. But hold on, there's a little detail we need to address. The inequality is < 0, not ≤ 0. This means that the fraction can't be equal to zero, and the denominator can't be zero either (because division by zero is undefined). Therefore, -4 and 1/2 are not included in the solution. We use parentheses to denote that the endpoints are not included in the interval. The final solution is expressed as an interval: (-4, 1/2). In set notation, it would be {x | -4 < x < 1/2}. It's important to understand the notation used to represent the solution. Parentheses indicate that the endpoints are not included, while square brackets would indicate that they are included. Because the inequality is strictly less than 0, we use parentheses.

So, to recap, the solution to the inequality (x+4) / (2x-1) < 0 is all 'x' values between -4 and 1/2, excluding -4 and 1/2. This is because at x = -4, the numerator becomes zero, and at x = 1/2, the denominator becomes zero, and division by zero is undefined. We've successfully navigated this inequality! You've learned how to find critical points, create intervals, and test those intervals to determine the solution. Keep practicing these types of problems, and you'll become a pro in no time! Remember, the key is to understand the concepts and the steps involved. You've got this!