Solving Absolute Value Inequality: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of absolute value inequalities. These might seem a little tricky at first, but don't worry, we'll break it down step by step. We'll not only solve the inequality but also learn how to represent the solution graphically. In this guide, we will solve the inequality $2|x+1|<4$ for $x$ and identify the graph of its solution. This is a common problem in algebra, and understanding how to solve it can be super helpful for your math journey. So, let's get started!

Understanding Absolute Value

First, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. It's always non-negative. For example, $|3| = 3$ and $|-3| = 3$. Think of it as stripping away the negative sign if there is one. The absolute value is a crucial concept in mathematics, representing the distance of a number from zero on the number line. This distance is always non-negative. Understanding this fundamental principle is key to grasping how absolute value inequalities work. For instance, both 3 and -3 have an absolute value of 3 because they are both 3 units away from zero. This concept extends to algebraic expressions as well. For an expression like $|x+1|$, we are considering the distance of the quantity $(x+1)$ from zero. Recognizing this distance aspect helps in translating the inequality into compound inequalities, which we will discuss later. The absolute value essentially provides a way to measure magnitude or size, irrespective of the sign. This is particularly useful in various mathematical and real-world applications, such as error analysis, where we are often concerned with the magnitude of the error rather than its direction. Moreover, the absolute value function plays a significant role in defining metrics and norms in higher mathematics, which are used to measure distances in abstract spaces. In the context of inequalities, the absolute value introduces a branching scenario: the expression inside the absolute value can be either positive or negative, but the absolute value ensures a non-negative result. This characteristic leads to the necessity of considering two separate cases when solving absolute value inequalities, which we will explore in detail as we tackle the problem at hand. By understanding the core concept of absolute value, we lay a solid foundation for handling more complex inequalities and related problems. Let's now move on to applying this understanding to solve our inequality.

Breaking Down the Inequality

The inequality we need to solve is $2|x+1|<4$. The first step is to isolate the absolute value expression. We can do this by dividing both sides of the inequality by 2. This gives us $|x+1|<2$. Now, here's the key idea: if the absolute value of something is less than 2, that something must be between -2 and 2. This means $x+1$ must be greater than -2 and less than 2. In mathematical terms, we can write this as a compound inequality: $-2 < x+1 < 2$. This is where things get interesting! We've transformed our absolute value inequality into a simpler compound inequality. This transformation is crucial because it allows us to deal with the two possibilities that arise from the absolute value: the expression inside the absolute value can be either positive or negative, but its distance from zero must still be less than 2. Breaking down the inequality in this way helps us understand the range of values that $x$ can take. The compound inequality $-2 < x+1 < 2$ essentially captures the constraint that the quantity $x+1$ must lie within 2 units of zero. This is a direct consequence of the definition of absolute value. When we encounter an absolute value inequality of the form $|expression| < a$, where $a$ is a positive number, it implies that $-a < expression < a$. This breakdown is not just a mathematical trick; it reflects the underlying concept of distance from zero. By understanding this, you can confidently tackle various absolute value inequalities. Next, we'll see how to solve this compound inequality to find the solution set for $x$. Remember, the goal is to isolate $x$ in the middle, so we'll be performing the same operations on all parts of the inequality to maintain the balance. Let's move on to the next step and solve for $x$!

Solving the Compound Inequality

So, we've got the compound inequality $-2 < x+1 < 2$. To solve for $x$, we need to isolate it in the middle. We can do this by subtracting 1 from all three parts of the inequality. This gives us $-2 - 1 < x+1 - 1 < 2 - 1$, which simplifies to $-3 < x < 1$. Ta-da! We've solved for $x$. This inequality tells us that $x$ must be greater than -3 and less than 1. In other words, $x$ lies between -3 and 1, but it cannot be equal to -3 or 1. This range of values is the solution to our original absolute value inequality. Solving the compound inequality is a straightforward process, but it's important to understand why we perform the same operation on all parts. The idea is to maintain the balance of the inequality. If we subtract 1 from the middle, we must also subtract 1 from the left and right sides to preserve the relationship. The result, $-3 < x < 1$, provides a clear and concise description of the solution set. It tells us that $x$ can take any value between -3 and 1, excluding the endpoints. This is a crucial distinction, and it's reflected in how we represent the solution graphically, which we'll discuss in the next section. The solution set is an interval on the number line, and it's defined by two critical points, -3 and 1. These points are the boundaries of the solution, and they determine the range of values that satisfy the original inequality. Now that we've found the solution, let's visualize it on a graph. This will help us understand the solution set in a more intuitive way. So, let's move on to the graphical representation of the solution.

Graphing the Solution

Now that we've found the solution $-3 < x < 1$, let's graph it on a number line. This will give us a visual representation of all the values of $x$ that satisfy the inequality. To graph this, we'll draw a number line and mark the points -3 and 1. Since $x$ is strictly greater than -3 and strictly less than 1 (not equal to), we'll use open circles at -3 and 1 to indicate that these points are not included in the solution. Then, we'll shade the region between -3 and 1 to show that all the values in this interval are solutions. The shaded region represents the set of all real numbers between -3 and 1. This visual representation is incredibly helpful because it allows us to quickly see the range of values that satisfy the absolute value inequality. Graphing the solution is not just a way to visualize the answer; it also helps in understanding the nature of the solution set. In this case, the solution is an open interval, meaning it does not include its endpoints. This is a direct consequence of the strict inequalities ($<$) in the solution $-3 < x < 1$. If the inequalities were non-strict ($\leq$), we would use closed circles (or brackets) to indicate that the endpoints are included in the solution. The graph provides a clear and intuitive way to communicate the solution to others. It's a universal language that transcends algebraic notation. Moreover, graphing the solution can help in identifying errors. If the graph doesn't match the algebraic solution, it's a sign that something went wrong in the process. Now that we've graphed the solution, we can confidently identify the correct graph from the given options. The graph should show an open interval between -3 and 1. This visual confirmation is a powerful tool in problem-solving. So, let's recap what we've done and then choose the correct answer.

Putting It All Together

Okay, let's recap what we've done to solve the absolute value inequality $2|x+1|<4$. First, we isolated the absolute value by dividing both sides by 2, giving us $|x+1|<2$. Then, we transformed this into a compound inequality: $-2 < x+1 < 2$. Next, we solved for $x$ by subtracting 1 from all parts of the inequality, which resulted in $-3 < x < 1$. Finally, we graphed the solution on a number line, showing an open interval between -3 and 1. Now, let's look at the options provided in the question. We need to choose the answer that gives both the correct solution and the correct graph. Based on our work, the correct solution is $-3 < x < 1$, which means $x$ is greater than -3 and less than 1. The graph should show an open interval between -3 and 1. Option C, "Solution: $x>-3$", is not the complete solution because it only gives one part of the inequality. Options A and B provide a different solution $x<-3$ or $x>1$ which is incorrect. The correct solution must capture the fact that $x$ is bounded both above and below. Putting it all together, we can confidently identify the correct answer. We've not only solved the inequality but also understood the underlying concepts and how to represent the solution graphically. This step-by-step approach is key to tackling similar problems in the future. Remember, the goal is not just to find the answer but to understand the process. By understanding the process, you can apply the same techniques to a wide range of problems. So, let's celebrate our success and move on to the next challenge! Absolute value inequalities might seem daunting at first, but with practice and a clear understanding of the steps involved, you can conquer them with ease.

Choosing the Correct Answer

Based on our comprehensive solution, we found that the solution to the absolute value inequality $2|x+1|<4$ is $-3 < x < 1$. The graph of this solution is a number line with an open interval between -3 and 1. This means that we have open circles at -3 and 1, and the region between these two points is shaded. Now, let's analyze the given options to choose the correct one.

• Option A: Solution: $x<-3$ or $x>1$; graph:

  This solution is incorrect because it represents the values of $x$ that are either less than -3 or greater than 1. This is the solution to the inequality $2|x+1|>4$, not $2|x+1|<4$. The graph would show two separate intervals, one extending to the left of -3 and the other extending to the right of 1.

• Option B: Solution: $x<-1$ or $x>3$; graph:

  This solution is also incorrect. It represents the values of $x$ that are either less than -1 or greater than 3. This is not the solution to our inequality. The graph would show two separate intervals, one extending to the left of -1 and the other extending to the right of 3.

• Option C: Solution: $x>-3$

  This solution is incomplete. While it correctly identifies that $x$ must be greater than -3, it misses the upper bound. It doesn't account for the fact that $x$ must also be less than 1. Therefore, this option is not the correct solution to the given inequality.

Therefore, from the options provided, none of them provide both the correct solution and the correct graph. The correct solution is $-3 < x < 1$, and the graph should show an open interval between -3 and 1. If we had to choose the closest one, option C partially captures the solution, but it's crucial to recognize that it's incomplete. The absence of a correct option highlights the importance of solving the problem independently and verifying the answer. In this case, we've done just that, and we know the correct solution and its graphical representation. This process underscores the value of understanding the underlying concepts and being able to apply them to solve problems accurately. So, remember to always double-check your work and ensure that your solution aligns with the problem's requirements. Let's keep practicing and mastering these skills!