Height Inequality: 8th Grade Class Heights Explained

Hey guys! Let's dive into a fun math problem today that involves figuring out how to write an inequality. Inequalities are super useful for representing a range of possibilities, and in this case, we’re looking at the heights of students in Susan's 8th-grade class. We'll break down the problem step by step, making sure everyone understands how to set up and interpret the inequality. Stick around, and you'll become an inequality pro in no time!

Understanding the Problem

So, here’s the deal: The average height of students in Susan's 8th-grade class is 56 inches. But here’s the twist – no student's height is more than 4 inches away from this average. This means the students' heights can vary, but they all fall within a certain range. Our mission is to figure out which inequality accurately shows all the possible heights ($h$) for any student in the class. Essentially, we need to capture both the minimum and maximum possible heights within this 4-inch range.

To really nail this, let's break down the key pieces of information. We know the average height is our central point, and the 4-inch difference acts as our boundary. Think of it like a buffer zone around the average. The height of any student can be up to 4 inches taller or shorter than the average. This "up to" part is super crucial because it hints at the type of inequality we'll need to use. Are we talking about a strict limit, or is there some wiggle room? Keep this in mind as we move forward.

We're not just plugging in numbers here; we're building a mathematical model of a real-world situation. This is what makes inequalities so powerful – they allow us to represent and solve problems where things aren't fixed but exist within a range. So, let's get ready to translate this word problem into a mathematical statement that captures the essence of the students' heights in Susan's class. We'll explore the different ways to express this relationship and choose the one that best fits the given conditions. Ready to jump into the nitty-gritty of setting up the inequality? Let's do it!

Setting up the Inequality

Alright, let's get down to business and figure out how to set up this inequality. Remember, we're trying to represent the possible heights ($h$) of students in Susan's class. The average height is 56 inches, and no student's height differs by more than 4 inches from this average. This means the student's height can be within 4 inches above or below the average. To capture this, we'll use an absolute value inequality. Absolute value is key here because we care about the distance from the average, not necessarily whether the height is above or below it.

The basic structure of our inequality will look something like this: | something | ≤ 4. The "≤ 4" part represents the maximum allowable difference of 4 inches. The "something" inside the absolute value bars will involve the student's height ($h$) and the average height (56 inches). The big question is, how do we arrange these to accurately show the difference?

Think of it this way: We want the absolute value of the difference between a student's height and the average height to be no more than 4 inches. This translates to | h - 56 | ≤ 4. This inequality is saying, “The absolute value of the difference between a student's height (h) and the average height (56) is less than or equal to 4.” This perfectly captures our problem's conditions! We're using the absolute value to ensure we consider both positive and negative differences (heights above and below the average). The "less than or equal to" (≤) is crucial because it includes the possibility that a student's height could be exactly 4 inches away from the average.

So, we've successfully translated the word problem into a concise mathematical statement. But what does this inequality really mean in terms of the students' heights? Let's dive deeper and explore how to interpret and solve this absolute value inequality to find the range of possible heights. We'll see how this seemingly simple inequality unlocks the boundaries within which all the students' heights must fall. Ready to break it down further? Let's go!

Solving the Inequality

Now that we've set up the inequality | h - 56 | ≤ 4, it's time to solve it and find the actual range of possible heights. Remember, absolute value inequalities can be a bit tricky because they involve two possibilities: the expression inside the absolute value can be positive or negative. To handle this, we need to split our single inequality into two separate inequalities.

The first inequality will represent the case where the expression inside the absolute value is positive or zero: h - 56 ≤ 4. This is pretty straightforward – it’s saying that the difference between a student's height and the average height is less than or equal to 4. To solve for h, we simply add 56 to both sides of the inequality: h ≤ 60. So, one boundary for the heights is 60 inches – no student can be taller than this.

But what about the students who are shorter than average? This is where the second inequality comes in. We need to consider the case where the expression inside the absolute value is negative. To do this, we flip the sign of the inequality and change the sign of the constant: h - 56 ≥ -4. Notice how the “≤” became a “≥” and the 4 became a -4. This is the crucial step in dealing with absolute value inequalities. Now, we solve for h just like before, by adding 56 to both sides: h ≥ 52. This tells us that the shortest a student can be is 52 inches.

So, we've found our two boundaries: h ≤ 60 and h ≥ 52. This means that any student in Susan's class must have a height between 52 inches and 60 inches, inclusive. We've successfully transformed an absolute value inequality into a clear range of possible values. We can even combine these two inequalities into a single compound inequality to represent the solution more compactly. Ready to see how it all fits together and what this range really means in the context of the problem? Let's keep going!

Interpreting the Solution

Okay, we've done the math, solved the inequality, and now we have a range of possible heights: h ≤ 60 and h ≥ 52. But what does this really mean for the students in Susan's class? It's super important to not just get the answer but to understand what it tells us about the real world.

These two inequalities tell us that the height of any student in the class must fall between 52 inches and 60 inches, including those endpoints. Think of it as a height window. No student can be shorter than 52 inches, and no student can be taller than 60 inches. The average height, 56 inches, sits right in the middle of this window, with 4 inches of wiggle room on either side.

We can also express this range using a compound inequality: 52 ≤ h ≤ 60. This is a concise way of saying the same thing – the height h is greater than or equal to 52 inches AND less than or equal to 60 inches. It's like a mathematical shorthand for our two separate inequalities.

So, to recap, we started with a word problem about the heights of students in Susan's class. We translated this problem into an absolute value inequality, solved it by splitting it into two separate inequalities, and then interpreted the solution to find the range of possible heights. We now know that every student in the class must be between 52 and 60 inches tall. This entire process highlights the power of inequalities in modeling real-world situations where we deal with ranges rather than fixed values. It's not just about finding the numbers; it's about understanding what those numbers represent.

Wrapping Up

Woohoo! We made it through the entire problem! We started with a word problem, transformed it into a mathematical inequality, solved it, and most importantly, understood what the solution means in the context of the problem. That’s the real magic of math – taking something from the real world and representing it with numbers and symbols.

We've seen how absolute value inequalities are incredibly useful for situations where we're dealing with distances from a central value. The 4-inch difference in heights was the key, and the absolute value allowed us to capture both the taller and shorter students relative to the average. Remember, the trick is to split the absolute value inequality into two separate inequalities, solve each one, and then interpret the results.

So, next time you encounter a problem that involves a range of possibilities, think about inequalities! They're your secret weapon for tackling those situations. And remember, math isn’t just about numbers; it's about understanding the world around us. You guys are awesome for sticking with it and conquering this inequality problem! Keep up the fantastic work, and I'll catch you in the next one!