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

Let's dive into the world of absolute value inequalities! In this article, we'll break down how to solve the inequality $2|9-x|-7 extgreater{=}-1$ algebraically. Don't worry, we'll take it one step at a time, making sure you understand the process thoroughly. We aim to provide a comprehensive, human-friendly explanation that will make solving these types of problems a breeze. So, grab your pen and paper, and let's get started!

Understanding Absolute Value Inequalities

Before we jump into solving the specific inequality, it's crucial to grasp the fundamental concept of absolute value. Think of absolute value as the distance a number is from zero, regardless of direction. For example, both |3| and |-3| equal 3 because both 3 and -3 are three units away from zero. When we deal with inequalities involving absolute values, we're essentially looking for a range of values that satisfy a certain distance condition. This means we often end up with two separate inequalities to solve, which we'll see in action shortly.

Absolute value inequalities can seem intimidating at first, but they're really just inequalities with an absolute value expression thrown into the mix. The key to solving them is to remember that the absolute value of a number is its distance from zero. This means that when we have an inequality like $|x| < 5$, we're looking for all the numbers that are less than 5 units away from zero. That includes numbers between -5 and 5. Similarly, if we have $|x| > 5$, we're looking for numbers that are more than 5 units away from zero, which means numbers less than -5 or greater than 5. Understanding this core concept is essential for tackling more complex absolute value inequalities.

The goal in solving any absolute value inequality is to isolate the absolute value expression first. This is very similar to how you would solve a regular algebraic equation. Once the absolute value part is by itself, you can then split the problem into two separate cases based on the definition of absolute value. Each of these cases will give you a different inequality to solve, and the solutions to both cases together will give you the complete solution to the original absolute value inequality. This step-by-step process is what makes these problems manageable and, dare I say, even fun once you get the hang of it! Remember, practice makes perfect, so don't be afraid to work through several examples to solidify your understanding.

Step-by-Step Solution for $2|9-x|-7 extgreater{=}-1$

Now, let's break down the solution to our inequality, $2|9-x|-7 extgreater{=}-1$. We'll go through each step methodically to ensure clarity.

1. Isolate the Absolute Value

The first step is to isolate the absolute value term. This means getting the $|9-x|$ part by itself on one side of the inequality. To do this, we'll start by adding 7 to both sides of the inequality:

$2|9-x|-7 + 7 extgreater{=}-1 + 7$

This simplifies to:

$2|9-x| extgreater{=} 6$

Next, we'll divide both sides by 2 to completely isolate the absolute value:

rac{2|9-x|}{2} extgreater{=} rac{6}{2}

Which gives us:

$|9-x| extgreater{=} 3$

Now that we have the absolute value isolated, we're ready to move on to the next crucial step: splitting the inequality into two separate cases.

2. Split the Inequality into Two Cases

The definition of absolute value tells us that $|9-x|$ can be either $(9-x)$ or $-(9-x)$, depending on whether the expression inside the absolute value is positive or negative. This is where we split the problem into two separate cases:

• Case 1: $9-x extgreater{=} 3$
• Case 2: $-(9-x) extgreater{=} 3$

In Case 1, we're considering the scenario where the expression inside the absolute value, $(9-x)$, is already non-negative. So, the absolute value doesn't change anything. In Case 2, we're dealing with the scenario where $(9-x)$ is negative. In this case, the absolute value makes it positive, which is why we multiply the expression by -1. Now we have two linear inequalities, which are much easier to solve.

3. Solve Case 1: $9-x extgreater{=} 3$

Let's solve the first case. We have the inequality:

$9-x extgreater{=} 3$

To isolate $x$, we'll subtract 9 from both sides:

$9-x-9 extgreater{=} 3-9$

This simplifies to:

$-x extgreater{=}-6$

Now, we need to get $x$ by itself. Since it's multiplied by -1, we'll multiply both sides by -1. Remember, when you multiply or divide an inequality by a negative number, you must flip the inequality sign:

$(-1)(-x) extless{=} (-1)(-6)$

Which gives us:

$x extless{=} 6$

So, the solution for Case 1 is $x extless{=} 6$. This means all values of $x$ that are less than or equal to 6 satisfy the first part of our inequality. Now, let's move on to Case 2 and see what solutions we get there.

4. Solve Case 2: $-(9-x) extgreater{=} 3$

Now, let's tackle the second case. We have the inequality:

$-(9-x) extgreater{=} 3$

First, we can distribute the negative sign:

$-9 + x extgreater{=} 3$

Next, add 9 to both sides to isolate $x$:

$-9 + x + 9 extgreater{=} 3 + 9$

This simplifies to:

$x extgreater{=} 12$

So, the solution for Case 2 is $x extgreater{=} 12$. This means all values of $x$ that are greater than or equal to 12 also satisfy our original inequality. Now that we've solved both cases, we're ready to combine the solutions.

5. Combine the Solutions

We've found that the solutions to our inequality are $x extless{=} 6$ from Case 1 and $x extgreater{=} 12$ from Case 2. This means that any value of $x$ that is either less than or equal to 6, or greater than or equal to 12, will satisfy the original inequality $2|9-x|-7 extgreater{=}-1$. We can express this solution in interval notation as:

$(-\infty, 6] extless{=} [12, \infty)$

This notation concisely represents all the numbers from negative infinity up to and including 6, and all the numbers from 12 up to positive infinity. This is the complete solution set for our absolute value inequality.

Visualizing the Solution

It's often helpful to visualize the solution on a number line. If we draw a number line, we can mark the points 6 and 12. Since our solution includes $x$ values less than or equal to 6, we'll draw a closed circle (or a filled-in dot) at 6 and shade the line to the left, indicating all numbers less than 6. Similarly, for $x$ values greater than or equal to 12, we'll draw a closed circle at 12 and shade the line to the right, indicating all numbers greater than 12. The number line clearly shows the two distinct intervals that make up our solution set.

Visualizing the solution helps to solidify the understanding of what the inequality means. It shows that there's a gap in the solution set between 6 and 12, which is a common characteristic of absolute value inequalities where the absolute value expression is greater than or equal to a positive number. The number line representation is a powerful tool for both understanding and communicating the solution to absolute value inequalities.

Conclusion

We've successfully solved the absolute value inequality $2|9-x|-7 extgreater{=}-1$ algebraically! We walked through each step, from isolating the absolute value to splitting the inequality into two cases, solving each case, and finally combining the solutions. Remember, the key to mastering these types of problems is practice. Work through various examples, and you'll become more comfortable with the process. Don't hesitate to review the steps we've outlined here whenever you encounter a challenging problem. And remember, absolute value inequalities, while they might seem tricky at first, are just another type of algebraic puzzle to be solved.

For further exploration and practice, you might find the resources available at Khan Academy helpful.