Finding Set Intersection Graphically: (-7, 8) ∩ [-5, 9]

Hey guys! Today, we're going to dive into a fun topic in mathematics: finding the intersection of sets using graphs. Specifically, we'll be looking at the sets (-7, 8) and [-5, 9]. If you're wondering what that funny symbol '∩' means, don't worry! It simply represents the intersection, which means we're looking for the elements that are common to both sets. Think of it as the overlap between two groups. So, let's break it down step by step and make it super clear.

Understanding Set Notation and Intervals

Before we jump into the graphical method, let's make sure we're all on the same page with set notation. You'll notice we have parentheses and brackets in our sets, like (-7, 8) and [-5, 9]. These symbols tell us whether or not the endpoints are included in the set.

• A parenthesis, like in (-7, 8), means that the endpoint is not included. So, (-7, 8) includes all numbers between -7 and 8, but not -7 and not 8 themselves. We can think of this as an open interval.
• A bracket, like in [-5, 9], means that the endpoint is included. So, [-5, 9] includes all numbers between -5 and 9, including -5 and including 9. This is a closed interval on the left and open on the right.

Understanding this difference is crucial because it affects the final intersection. If an endpoint is not included in one of the sets, it cannot be part of the intersection. Basically, to be in the intersection, a number has to be in both sets. We're laying the groundwork here, folks, so make sure you've got this down. It's like understanding the rules of a game before you start playing – essential for success!

Visualizing Intervals on a Number Line

Now that we understand the notation, let's visualize these intervals on a number line. This is where the graphical method starts to take shape. Grab a piece of paper (or your favorite digital drawing tool) and draw a horizontal line. This is our number line. Mark the relevant numbers: -7, -5, 8, and 9. These are the boundaries of our sets.

For the interval (-7, 8), we'll draw an open circle at -7 and another open circle at 8. Then, we'll draw a line connecting these circles. The open circles indicate that -7 and 8 are not included in the set. Think of it like a fence with holes at the endpoints – you can't stand right on the hole!

For the interval [-5, 9], we'll draw a closed circle (or a filled-in circle) at -5 and an open circle at 9. Then, we'll draw a line connecting these circles. The closed circle at -5 indicates that -5 is included in the set. It's like a solid part of the fence where you can stand. The open circle at 9 indicates 9 is not included. This visual representation is key because it helps us see where the sets overlap. It's like having a map that shows you the terrain you're navigating – much easier than trying to figure it out in your head!

Graphing the Sets (-7, 8) and [-5, 9]

Alright, let's put those number line skills into action! We're going to graph both sets, (-7, 8) and [-5, 9], on the same number line. This is where the magic happens, guys. This visual representation will make finding the intersection super clear.

First, draw your number line and mark the key points: -7, -5, 8, and 9. Now, let's graph (-7, 8). As we discussed, we'll use open circles at -7 and 8 and connect them with a line. You can even shade the line to represent all the numbers in between. This shaded line represents all real numbers strictly between -7 and 8.

Next, let's graph [-5, 9]. This time, we'll use a closed circle at -5 (because -5 is included) and an open circle at 9 (because 9 is not included). Connect these circles with a line and shade it. This shaded line represents all real numbers from -5 (inclusive) up to 9 (exclusive).

Now, look closely at your number line. Where do the two shaded regions overlap? This overlapping region is the intersection of the two sets! It's like finding the common ground between two territories. This visual method makes it so much easier to grasp than just staring at the numbers and symbols. You can see the intersection!

Identifying the Overlapping Region

Okay, we've got our graphs, and we can see the overlap. But how do we express that overlap as a set? This is the crucial step in finding the intersection. The overlapping region represents the numbers that are present in both sets. It's like the sweet spot where the Venn diagram circles intersect, if you're familiar with those.

Look at the left boundary of the overlap. It starts at -5. Now, is -5 included in the intersection? Yes, it is! Because -5 is included in the set [-5, 9], and while it's not included in (-7,8), that set only goes up to -7, meaning -5 is within that range. So, we'll use a bracket here: [.

Now, let's look at the right boundary of the overlap. It ends at 8. Is 8 included in the intersection? No, it's not! Because 8 is not included in the set (-7, 8) (remember the parenthesis?). So, we'll use a parenthesis here: ).

Putting it all together, the overlapping region, and therefore the intersection of the sets, is [-5, 8). We've visually identified the common ground and expressed it in proper set notation. High five!

Determining the Intersection Set

So, we've graphed our sets, identified the overlapping region, and now we need to write down the actual intersection set. This is where we translate our visual understanding into mathematical notation. We're taking the picture we've created and turning it into words, or in this case, numbers and symbols!

Looking at our graph, the overlapping region starts at -5 (inclusive) and goes all the way up to 8 (exclusive). Remember, the closed circle at -5 means it's included, and the open circle at 8 means it's not. It's like saying,