Solving Absolute Value Inequality $2|x+7|-4 \geq 0$ In Set-Builder Notation

Hey guys! Today, we're going to dive into solving an absolute value inequality and expressing the solution using set-builder notation. This type of problem might seem a bit tricky at first, but don't worry, we'll break it down step by step so it becomes super clear. Inequalities involving absolute values are a fundamental topic in algebra, and mastering them will definitely boost your problem-solving skills. So, let's jump right into it and see how we can tackle this!

Understanding Absolute Value Inequalities

Before we get into the specific problem, let's quickly recap what absolute value means and how it affects inequalities. 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 $|-3| = 3$. When we deal with inequalities involving absolute values, we need to consider two cases: one where the expression inside the absolute value is positive or zero, and another where it’s negative. This is because the absolute value strips away the negative sign, so we have to account for both possibilities to find the complete solution.

When you first encounter absolute value inequalities, it's easy to feel a bit overwhelmed, but trust me, with a systematic approach, they become quite manageable. The key is to remember to split the problem into these two cases and handle each one separately. Make sure you understand this concept really well because it forms the basis for solving these types of problems. We're going to apply this understanding directly to our problem, so let's keep this in mind as we move forward. Grasping the fundamental principle of splitting the absolute value into its positive and negative scenarios is crucial for successfully solving these inequalities. Keep practicing, and soon, you'll find these problems become second nature!

Step-by-Step Solution

Our inequality is $2|x+7|-4 ext{ }\geq 0$. Our goal is to isolate the absolute value expression first.

1. Isolate the absolute value term: Add 4 to both sides of the inequality:

   $2|x+7| ext{ }\geq 4$

   Divide both sides by 2:

   $|x+7| ext{ }\geq 2$

2. Split into two cases: Now, we'll consider two cases based on the definition of absolute value.

   • Case 1: The expression inside the absolute value is positive or zero:

     $x+7 ext{ }\geq 2$

   • Case 2: The expression inside the absolute value is negative:

     $-(x+7) ext{ }\geq 2$

3. Solve each case:

   • Case 1: Subtract 7 from both sides:

     $x ext{ }\geq 2 - 7$

     $x ext{ }\geq -5$

   • Case 2: Distribute the negative sign:

     $-x - 7 ext{ }\geq 2$

     Add 7 to both sides:

     $-x ext{ }\geq 9$

     Multiply both sides by -1 (and remember to flip the inequality sign):

     $x ext{ }\leq -9$

4. Combine the solutions: We found two solution sets: $x ext{ }\geq -5$ and $x ext{ }\leq -9$. This means our solution includes all $x$ values that are either greater than or equal to -5 OR less than or equal to -9.

Expressing the Solution in Set-Builder Notation

Set-builder notation is a way to describe a set by specifying the properties its elements must satisfy. It generally looks like this: ${x ext{ }| ext{ } condition}$.

In our case, the condition is that $x$ must be less than or equal to -9 or greater than or equal to -5. So, we can write our solution set in set-builder notation as:

${x ext{ }| ext{ } x ext{ }\leq -9 ext{ or } x ext{ }\geq -5}$

And that's it! We've successfully solved the inequality and expressed the solution in set-builder notation.

Common Mistakes to Avoid

When working with absolute value inequalities, there are a few common mistakes that students often make. Recognizing these pitfalls can save you a lot of headaches. One frequent error is forgetting to consider both the positive and negative cases of the absolute value expression. Remember, the absolute value of a number is its distance from zero, so both positive and negative values inside the absolute value can lead to the same result. Another mistake is not flipping the inequality sign when multiplying or dividing by a negative number. This is a crucial step in solving inequalities, and overlooking it will lead to an incorrect solution. Always double-check whether you need to flip the sign.

Additionally, some students struggle with combining the solutions from the two cases correctly. It's essential to understand whether the solutions should be combined using “and” or “or.” In our example, we used “or” because the solution includes values that satisfy either $x ext{ }\leq -9$ or $x ext{ }\geq -5$. A clear understanding of the logical connectives is crucial for expressing the solution set accurately. By being mindful of these common mistakes, you can increase your accuracy and confidence in solving absolute value inequalities. Practice identifying these errors in your own work and in examples, and you'll soon find yourself mastering these types of problems.

Practice Problems

To really nail this concept, it's essential to practice. Here are a couple of similar problems you can try:

1. Solve $|2x - 1| ext{ }\geq 5$ and express the solution in set-builder notation.
2. Solve $3|x + 2| - 6 ext{ }\geq 0$ and express the solution in set-builder notation.

Work through these problems using the steps we've discussed. Compare your solutions with friends or check them online to make sure you're on the right track. The more you practice, the more comfortable you'll become with these types of problems.

Real-World Applications

You might be wondering, “Where would I ever use this in real life?” Well, absolute value inequalities pop up in various fields, especially in situations where you're dealing with tolerances or acceptable ranges. For instance, in engineering, when designing parts, there might be an acceptable range of measurements. If a part's measurement falls outside this range, it's considered defective. These ranges can often be expressed using absolute value inequalities. Similarly, in statistics, absolute value inequalities are used to define confidence intervals and margins of error.

Consider a scenario in manufacturing where a machine produces bolts with a specified diameter. The acceptable diameter might be $10 ext{ mm} ext{ }\pm 0.1 ext{ mm}$. This means the diameter can be anywhere between 9.9 mm and 10.1 mm. We can represent this situation using an absolute value inequality: $|d - 10| ext{ }\leq 0.1$, where $d$ is the diameter of the bolt. Solving this inequality helps determine the range of acceptable bolt diameters. Understanding these practical applications not only makes the math more relevant but also shows how valuable these skills are in various professions. So, the next time you encounter an absolute value inequality, think about the real-world scenarios where it might be applied!

Conclusion

So, there you have it! We've walked through how to solve the inequality $2|x+7|-4 ext{ }\geq 0$ and express the solution in set-builder notation. Remember, the key is to isolate the absolute value, split the problem into two cases, solve each case separately, and then combine the solutions. Set-builder notation is just a fancy way of describing the solution set precisely. Practice makes perfect, so keep at it, and you'll become a pro at solving these types of problems. Keep exploring and happy solving!