Solving Absolute Value Inequality: |2w-7|+3 < 5

Hey guys! Today, we're diving into solving an absolute value inequality. Specifically, we're tackling the problem: $|2w - 7| + 3 < 5$. Don't worry, it might look a bit intimidating at first, but we'll break it down step-by-step so it's super easy to understand. We'll go through the process of isolating the absolute value, setting up the compound inequality, and then solving for w. By the end of this article, you'll be a pro at handling these types of problems. So, let's jump right in and get started!

Understanding Absolute Value Inequalities

Before we jump into solving our specific problem, it's essential to understand what absolute value inequalities are all about. An absolute value represents the distance of a number from zero on the number line. This means it's always non-negative. For example, both |3| and |-3| equal 3 because both 3 and -3 are three units away from zero. When we deal with absolute value inequalities, we're looking at ranges of values that satisfy a certain distance condition. Inequalities involving absolute values can be a little tricky because we have to consider two scenarios: the expression inside the absolute value can be either positive or negative. This leads to two separate inequalities that we need to solve. Thinking about the number line and visualizing distances can be really helpful when working with these types of problems. Remember, the key is to isolate the absolute value expression first, and then split the problem into two cases. We'll see exactly how this works as we solve our example problem. So, keep this in mind as we move forward, and you'll find that absolute value inequalities are not as daunting as they might seem.

Isolating the Absolute Value

The first crucial step in solving any absolute value inequality is to isolate the absolute value expression. This means we want to get the part with the absolute value, in our case $|2w - 7|$, all by itself on one side of the inequality. Looking at our problem, $|2w - 7| + 3 < 5$, we see that there's a '+ 3' hanging out on the same side as the absolute value. To get rid of it, we need to perform the inverse operation, which is subtraction. So, we'll subtract 3 from both sides of the inequality. This gives us: $|2w - 7| < 5 - 3$, which simplifies to $|2w - 7| < 2$. Now we've successfully isolated the absolute value! This step is super important because it sets us up to handle the two possible cases that arise from the absolute value. Remember, the absolute value of something is its distance from zero, so $|2w - 7| < 2$ means that the expression inside the absolute value, $2w - 7$, is less than 2 units away from zero. This can be either positive or negative, which is why we need to consider both cases. With the absolute value isolated, we're ready to split the inequality into two separate cases and solve each one. So, let's move on to the next step!

Setting up the Compound Inequality

Now that we've isolated the absolute value, the next step is to set up the compound inequality. This is where we split our original absolute value inequality into two separate inequalities that we can solve individually. Remember, the absolute value $|2w - 7|$ represents the distance of the expression $2w - 7$ from zero. So, when we say $|2w - 7| < 2$, we're saying that this distance is less than 2. This means that $2w - 7$ must be between -2 and 2. In other words, it must be greater than -2 and less than 2. This gives us our two inequalities:

1. $2w - 7 < 2$
2. $2w - 7 > -2$

Notice how we've taken the expression inside the absolute value, $2w - 7$, and set it less than 2 (the original inequality) and also greater than -2. This is the key to handling absolute value inequalities. We're essentially saying that the expression inside the absolute value can be within 2 units of zero in either direction. Writing out these two inequalities is crucial because it allows us to solve for w in both scenarios. Once we've solved each inequality, we'll combine the solutions to get the final answer. So, with our compound inequality set up, we're ready to move on to the exciting part: solving for w!

Solving the Compound Inequality

Alright, let's dive into solving the compound inequality we set up in the previous section. We have two inequalities to tackle:

1. $2w - 7 < 2$
2. $2w - 7 > -2$

We'll solve each one separately, just like we would with any regular inequality. Let's start with the first one, $2w - 7 < 2$. To isolate w, we first need to get rid of the -7. We do this by adding 7 to both sides of the inequality: $2w - 7 + 7 < 2 + 7$, which simplifies to $2w < 9$. Now, to get w completely by itself, we divide both sides by 2: $\frac{2w}{2} < \frac{9}{2}$, which gives us $w < 4.5$. So, that's the solution for our first inequality! Now, let's move on to the second one, $2w - 7 > -2$. Again, we start by adding 7 to both sides: $2w - 7 + 7 > -2 + 7$, which simplifies to $2w > 5$. Then, we divide both sides by 2: $\frac{2w}{2} > \frac{5}{2}$, which gives us $w > 2.5$. Fantastic! We've solved both inequalities. We found that w must be less than 4.5 and greater than 2.5. Now, all that's left is to put these two solutions together to get the final answer.

Combining the Solutions

Now that we've solved both inequalities, it's time to combine the solutions to find the overall solution to our original absolute value inequality. We found that:

1. $w < 4.5$
2. $w > 2.5$

These two inequalities tell us that w must be both less than 4.5 and greater than 2.5. In other words, w lies between 2.5 and 4.5. We can express this as a compound inequality: $2.5 < w < 4.5$. This means that any value of w that falls within this range will satisfy the original inequality, $|2w - 7| + 3 < 5$. Think about it on a number line: we have an open circle at 2.5 (because w is strictly greater than 2.5) and another open circle at 4.5 (because w is strictly less than 4.5), and the solution is all the numbers in between. This gives us a clear and concise way to represent the solution. So, the final solution to our problem is that w is between 2.5 and 4.5. We've successfully navigated through the steps of solving an absolute value inequality, from isolating the absolute value to setting up and solving the compound inequality, and finally, combining the solutions. Great job, guys!

Final Answer

So, after working through all the steps, we've arrived at the final answer for the inequality $|2w - 7| + 3 < 5$. We found that the solution is $2.5 < w < 4.5$. This means that w can be any number between 2.5 and 4.5, but it cannot be equal to 2.5 or 4.5. To recap, we first isolated the absolute value expression by subtracting 3 from both sides, giving us $|2w - 7| < 2$. Then, we set up a compound inequality by considering both cases: when the expression inside the absolute value is positive and when it's negative. This led us to two inequalities: $2w - 7 < 2$ and $2w - 7 > -2$. We solved each inequality separately, finding $w < 4.5$ and $w > 2.5$. Finally, we combined these solutions to get our final answer: $2.5 < w < 4.5$. This whole process demonstrates the systematic approach to solving absolute value inequalities. Remember to isolate, split into cases, solve, and combine! You've now got another tool in your math toolbox. Keep practicing, and you'll become even more confident in tackling these types of problems. Well done, everyone!