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

Hey math enthusiasts! Today, we're diving into the world of absolute value inequalities. Specifically, we're going to break down how to solve an inequality like the one you presented: $\frac{2}{5}|5x + 10| - 14 > -6$. Don't worry if it looks a bit intimidating at first – we'll go through it step by step, making sure everything is super clear and easy to follow. Get ready to flex those math muscles! This guide provides a comprehensive approach to tackling absolute value inequalities. We'll explore the core concepts, learn the rules, and practice solving a variety of problems to solidify your understanding. By the end, you'll be able to confidently solve this type of inequality and apply these skills to similar problems.

Understanding Absolute Value

Before we jump into the inequality itself, let's make sure we're all on the same page about absolute value. Absolute value is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. It's always a non-negative value. Think of it like this: regardless of whether a number is positive or negative, its absolute value is always positive or zero. For example, the absolute value of 5, written as |5|, is 5. And the absolute value of -5, written as |-5|, is also 5. Guys, it's all about distance, and distance is never negative! Understanding this concept is crucial for solving absolute value inequalities correctly. When we deal with absolute values in inequalities, it means we're considering two possibilities: the expression inside the absolute value can be positive or negative, and we need to account for both scenarios. This duality is what makes solving absolute value inequalities a bit more involved than solving regular inequalities. But fear not, we'll break it down into manageable steps.

So, the absolute value is a fundamental concept in mathematics that measures the distance of a number from zero on a number line. It's always a non-negative value, making it a critical aspect of understanding inequalities, so we must understand the core principle. For example, |3| is 3, and |-3| is also 3 because both numbers are three units away from zero. This understanding is crucial for correctly interpreting and solving absolute value inequalities. In the context of the initial problem, the expression inside the absolute value, |5x + 10|, signifies the distance of the quantity (5x + 10) from zero. When solving the inequality, we consider two cases: when the expression inside the absolute value is positive or zero and when it's negative. This is what differentiates solving absolute value inequalities from solving standard inequalities, demanding that we account for both potential scenarios to arrive at the correct solution. Remember, the absolute value gives us the distance, so the solution is where the expression is greater or less than a value, no matter its sign.

Step-by-Step Solution

Alright, let's get down to business and solve $\frac{2}{5}|5x + 10| - 14 > -6$. Here's how we break it down, step by step, making sure every move is crystal clear:

1. Isolate the Absolute Value Term: Our first goal is to get the absolute value expression by itself on one side of the inequality. To do this, we need to get rid of the -14 and the $\frac{2}{5}$. First, let's deal with the -14. Add 14 to both sides of the inequality: $\frac{2}{5}|5x + 10| - 14 + 14 > -6 + 14$. This simplifies to $\frac{2}{5}|5x + 10| > 8$.
2. Isolate the Absolute Value: Now, we need to get rid of the $\frac{2}{5}$. Multiply both sides of the inequality by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$: $\frac{5}{2} \cdot \frac{2}{5}|5x + 10| > 8 \cdot \frac{5}{2}$. This simplifies to $|5x + 10| > 20$. We've now successfully isolated the absolute value term.
3. Create Two Inequalities: Because of the nature of absolute value, we now have to consider two separate cases. Remember, the absolute value of an expression is its distance from zero, so we need to account for both positive and negative values. Our inequality $|5x + 10| > 20$ means that the expression inside the absolute value, (5x + 10), is either greater than 20 or less than -20. So, we create two separate inequalities:
   • Case 1: $5x + 10 > 20$
   • Case 2: $5x + 10 < -20$
4. Solve Each Inequality: Now, let's solve each of these inequalities separately.
   • Case 1: $5x + 10 > 20$: Subtract 10 from both sides: $5x > 10$. Then, divide both sides by 5: $x > 2$.
   • Case 2: $5x + 10 < -20$: Subtract 10 from both sides: $5x < -30$. Then, divide both sides by 5: $x < -6$.
5. Write the Solution: The solution to the original absolute value inequality is the combination of the solutions from both cases. Therefore, the solution is $x < -6$ or $x > 2$. This means that any value of x less than -6 or greater than 2 will satisfy the original inequality. Good job, guys! You've successfully navigated the absolute value inequality.

We started with a complex-looking inequality, and through step-by-step simplification, we've broken it down to its core elements. By understanding the properties of absolute value and systematically applying algebraic operations, we were able to isolate the variable and find its range of valid values. Remember that practice is key, and as you work through more examples, you'll become more comfortable with the process. The core of this type of problem involves understanding absolute value, isolating the absolute value term, and creating two inequalities to solve. Each step is crucial, and paying attention to detail is key to accurate solutions. Congratulations on conquering this problem – you're well on your way to mastering absolute value inequalities!

Graphing the Solution

Graphing the solution on a number line can help visualize the solution set. For our solution, $x < -6$ or $x > 2$, we'll have an open circle at -6 and an open circle at 2, with the line shaded to the left of -6 and to the right of 2. The open circles indicate that -6 and 2 are not included in the solution set. This graphical representation provides a clear visual of all the values of x that satisfy the inequality. The number line will have an open circle at -6, shading to the left, and an open circle at 2, shading to the right. This represents all the values of x that satisfy the inequality, making the solution clear. It's a great way to check your work and ensure you understand the solution set. Always remember that graphing your solution can give you a different perspective of the problem, and may allow you to detect potential errors. Additionally, graphing will help you understand the concept of inequalities better.

So, as you can see, visualizing the solution on a number line gives you a clear picture of all the values of 'x' that meet the criteria of the original inequality. You can see how the solution spans two separate intervals, visually representing the values that satisfy the condition either less than -6 or greater than 2. This visual aid is incredibly helpful for understanding the solution set and ensuring you haven't made any mistakes. Remember that a number line helps in identifying the solution set for your inequality. Also, you can identify how the end numbers are included or excluded in your answers. Using the method of graphing the solution can not only give you a different perspective, but can also help you identify mistakes.

Practice Problems

Want to sharpen your skills? Here are a few practice problems to test your understanding. Try these on your own, and then check your answers. Remember, practice makes perfect!

1. Solve: $|2x - 4| \leq 6$
2. Solve: $\frac{1}{3}|x + 5| - 2 > 1$
3. Solve: $|3x + 9| < 12$

Answers:

1. -1 ≤ x ≤ 5
2. x < -14 or x > 4
3. -7 < x < 1

These practice problems provide a great opportunity to apply what you've learned. They'll help you build confidence and solidify your ability to solve absolute value inequalities. Remember, the key is to follow the same steps we've discussed: isolate the absolute value, create two inequalities, and solve each one. The answers are provided so you can check your work and learn from any mistakes. Don't be afraid to try different problems, as this is how you'll truly master the subject.

Keep practicing, and you'll be a pro in no time! Solving absolute value inequalities is all about understanding the basics and practicing regularly. Don't worry if it seems tough at first; with enough practice, you'll become confident in solving these types of problems. Remember to always double-check your work and to understand the underlying principles. Good luck, and keep up the great work, everyone! The key here is repetition and practice. The more problems you solve, the more comfortable you'll get with the process. Keep at it, and you'll do great. Good luck on your journey to master these types of problems, and remember to check your answers and understand where you might have gone wrong.

Conclusion

Congratulations, you've reached the end! We've successfully navigated through solving absolute value inequalities, from understanding the basic concept of absolute value to solving a specific example and providing practice problems. Remember, the key takeaways are to isolate the absolute value term and solve two separate inequalities. Guys, keep practicing, and you'll be well on your way to mastering these problems. Each step is crucial, and understanding why we do what we do is essential for long-term retention and mastery. This journey isn't just about finding the right answers; it's about developing your critical thinking and problem-solving skills. So keep learning, practicing, and challenging yourselves. The more you work with these inequalities, the more comfortable you'll become, and the more confident you'll be in your math abilities. Keep up the excellent work! And remember, math is a journey, not a destination, so enjoy the ride, and keep exploring the fascinating world of numbers and equations.