Solving Absolute Value Inequalities: A Comprehensive Guide

Hey there, math enthusiasts! Today, we're diving into the world of absolute value inequalities. Don't worry, it's not as scary as it sounds! We'll break down the process step-by-step, making sure you understand how to solve these problems. Let's get started by tackling the absolute value inequality $\frac{4|x+7|}{3} \lt 8$. This guide will walk you through the problem, providing a clear and concise explanation to help you master this type of problem. Understanding absolute value inequalities is a fundamental skill in algebra, opening the door to more complex mathematical concepts. So, grab your pencils, and let's begin our mathematical journey!

Understanding Absolute Value

Before we jump into the inequality, 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 a non-negative value. For example, $|3| = 3$ and $|-3| = 3$. The absolute value removes the sign, leaving only the magnitude. Now, think about what that means for inequalities. When we have an absolute value inequality, we're essentially saying the distance of something from zero must be less than or greater than a certain value. This often leads to two separate cases we need to consider. For $\frac{4|x+7|}{3} \lt 8$, the expression inside the absolute value bars, $|x+7|$, represents the distance between $x$ and $-7$ on the number line. So, we are saying that four-thirds of this distance must be less than 8. Understanding this visual representation is key to grasping the concept and helps in solving the problem accurately. Keep in mind that absolute value problems always have two possible solutions. We must consider both to find the complete solution set. Let's move on to the first step of solving our inequality, where we'll apply this understanding to our specific problem.

Isolating the Absolute Value Expression

Our first step in solving $\frac{4|x+7|}{3} \lt 8$ is to isolate the absolute value expression. Think of it like this: we want to get the $|x+7|$ all by itself on one side of the inequality. To do this, we need to get rid of the fraction $\frac{4}{3}$. How do we do that? Well, we multiply both sides of the inequality by $\frac{3}{4}$. This will cancel out the fraction on the left side, leaving us with just the absolute value expression. So, we have

$\frac{3}{4} \cdot \frac{4|x+7|}{3} \lt 8 \cdot \frac{3}{4}$

This simplifies to

$|x+7| \lt 6$

Great! Now we have the absolute value expression isolated. We've successfully simplified the given inequality to a more manageable form. This is a crucial step because it allows us to focus directly on the absolute value and the conditions it implies. Now, we are ready to solve the absolute value inequality. The inequality $|x+7| \lt 6$ tells us that the distance between $x$ and $-7$ is less than 6. This means $x$ must lie within a certain range on the number line. We'll explore this range and find the solution set in the next steps. Remember that correctly isolating the absolute value is fundamental to finding the accurate solution, so always pay attention to this step, ensuring you perform the algebraic operations correctly. By removing the fraction, we've set the stage for the next critical phase of finding the solutions, which requires careful consideration of the properties of absolute values.

Solving the Absolute Value Inequality

Now we have $|x+7| \lt 6$. This means the expression inside the absolute value bars, $x+7$, must be within 6 units of zero on the number line. To solve this, we need to consider two separate cases. The first case is when $x+7$ is positive, and the second is when $x+7$ is negative. For the first case, we write

$x+7 \lt 6$

Subtracting 7 from both sides gives us

$x \lt -1$

This tells us that $x$ must be less than $-1$. But we also have the second case when $x+7$ is negative. In this case, we write

$-(x+7) \lt 6$

Multiplying both sides by $-1$ (and remembering to flip the inequality sign), we get

$x+7 \gt -6$

Subtracting 7 from both sides gives us

$x \gt -13$

So, $x$ must be greater than $-13$. Now, we have two conditions: $x \lt -1$ and $x \gt -13$. Combining these two conditions, we get $-13 \lt x \lt -1$. This is the solution to the inequality. Always remember to handle the absolute value in terms of two inequalities to cover all possible conditions. It's easy to overlook one of the conditions, so take your time, be meticulous, and double-check your work. Once you have these two conditions, you can see the values that $x$ can take. Now that we have both conditions, we can define the range of values of $x$ and find the solution.

Final Solution

So, after all that work, we have the solution to $\frac{4|x+7|}{3} \lt 8$ as $-13 \lt x \lt -1$. This means that $x$ can be any number between $-13$ and $-1$, but it cannot equal either of those values. On the number line, this would be represented as an open interval from $-13$ to $-1$, often written as $(-13, -1)$. In the context of your original question, we can write the final answer like this:

$\frac{4|x+7|}{3} \lt 8$

$x \lt -1$

$x \gt -13$

So, to answer your initial question, $x \lt -1$ and $x \gt -13$. You can put these values into your homework or practice problems. Keep practicing, and you'll become a master of absolute value inequalities in no time. Remember, understanding each step is key to solving these problems successfully. Don't hesitate to review the steps or practice more examples to solidify your knowledge. The more you practice, the easier it will become. Math is all about consistency, and you've got this!

Conclusion

In conclusion, we've walked through solving the absolute value inequality $\frac{4|x+7|}{3} \lt 8$. We started with the basics of absolute value, isolated the absolute value expression, and then solved the resulting inequality by considering two cases. This approach helps us find the range of values that satisfy the inequality. Remember, practice makes perfect. Keep practicing, and you'll master these concepts. Understanding how to solve these absolute value inequalities is a valuable skill that will come in handy in more advanced math courses. Keep exploring, and stay curious!