Rozwiąż Nierówności: Graficzne Zbiory Punktów

Dec 9, 2025 by Admin 46 views

Hey guys! Ever stared at a math problem and felt like you needed a decoder ring to figure it out? I totally get it. Sometimes, especially with coordinate geometry and inequalities, it can feel like a puzzle. But don't sweat it! Today, we're diving deep into how to visually represent sets of points on a coordinate plane based on given conditions. We'll be tackling a specific problem that involves defining ranges for both x and y coordinates, and trust me, by the end of this, you'll be a pro at plotting these bad boys.

This is all about understanding how to break down complex conditions into simple, manageable steps. We're going to look at how the union symbol '∪' works and what those interval notations mean. Remember, math is like a language, and once you learn the grammar, it all starts to make sense. So, grab your graphing paper, your favorite pens, and let's get this done!

Understanding the Basics: Intervals and Unions

Alright, let's get our heads around the building blocks of this problem. We're dealing with intervals, which are just fancy words for a range of numbers on a number line. You'll see notations like [-3;-1] or (1;2). The square brackets [] mean the endpoints are included in the set (so, -3 and -1 are part of the set), while the parentheses () mean the endpoints are not included (so, 1 and 2 are not part of the set, but numbers very close to them are).

Now, the union symbol '∪' is super important here. Think of it as saying "or". When you see A ∪ B, it means you include all the numbers that are in set A, or in set B, or in both. It's like combining two separate lists of numbers into one big list. For our problem, the condition for x is x∈[-3;-1] ∪ (1;2). This means our x values can be anywhere between -3 and -1 (inclusive), or they can be anywhere between 1 and 2 (exclusive of 1 and 2). So, we have a chunk of the number line from -3 to -1, and then another separate chunk from just after 1 up to just before 2.

When we apply this to a 2D coordinate plane, we're not just looking at a line anymore. We're looking at areas. If x is in a certain interval, it means any x value within that interval is valid. If y is in a certain interval, any y value within that interval is valid. When we combine conditions for x and y, we're essentially drawing rectangles or strips on the graph. The union ∪ tells us we'll have multiple, possibly disconnected, regions on our graph.

Let's break down the x condition: x∈[-3;-1] ∪ (1;2). On the x-axis, this looks like:

A solid line segment from -3 to -1, including both endpoints.
A dashed line segment from 1 to 2, excluding both endpoints.

So, any point on our graph must have an x-coordinate that falls within either of these two ranges. This is our "base" region, and we'll be stacking the y conditions on top of this.

Plotting the Regions: Case by Case

Now for the fun part – let's actually plot these points! We'll take the condition for x that we just discussed and combine it with each of the conditions for y given in parts a), b), and c).

Part a) y ∈ [-2;-3]

First off, let's address the y condition in part a): y ∈ [-2;-3]. Wait a second, guys, does that look right? Usually, intervals are written with the smaller number first. So, [-2;-3] is a bit unusual. It's likely a typo and should probably be y ∈ [-3;-2] or y ∈ [-2; -3] if we interpret the order as given. Assuming it's meant to be a valid interval, and conventionally written from smallest to largest, let's assume it means y is between -3 and -2, inclusive. If the intention was literally [-2;-3], it would represent an empty set since -2 is greater than -3. For the sake of demonstrating the plotting, let's assume the intended interval was y ∈ [-3; -2]. This means our y values must be between -3 and -2, and both -3 and -2 are included.

Now, we combine this with our x condition: x∈[-3;-1] ∪ (1;2). We need points (x,y) where x is in [-3;-1] OR (1;2), AND y is in [-3;-2].

This creates two rectangular regions on our graph:

Region 1: The x values are from -3 to -1 (inclusive), and the y values are from -3 to -2 (inclusive). This forms a rectangle with corners at (-3, -3), (-1, -3), (-1, -2), and (-3, -2). All points within and on the boundary of this rectangle are included.
Region 2: The x values are from 1 to 2 (exclusive), and the y values are from -3 to -2 (inclusive). This forms another rectangle. However, since the x values exclude 1 and 2, the vertical sides of this rectangle at x=1 and x=2 will be dashed lines, not solid. The corners would conceptually be at (1, -3), (2, -3), (2, -2), and (1, -2). All points strictly between x=1 and x=2, and between y=-3 and y=-2 (inclusive) are included.

Visually, you'll have a solid rectangle in the bottom-left quadrant (and extending into the bottom-right) and a separate, dashed-boundary rectangle further to the right, also in the lower quadrants.

Part b) y ∈ [-3; -1] ∪ (1;3]

Okay, moving on to part b)! This one is a bit more complex because both x and y have conditions involving unions. Remember our x condition remains the same: x∈[-3;-1] ∪ (1;2).

The new y condition is y ∈ [-3; -1] ∪ (1;3]. This means y can be:

Between -3 and -1 (inclusive), OR
Between 1 and 3 (inclusive of 3, but exclusive of 1).

Now, we need to combine these with our x intervals. When we have (x condition) AND (y condition), and both conditions involve unions, we essentially create a grid of possibilities. We need to consider each combination of x intervals and y intervals.

Let's break it down:

Combination 1: x ∈ [-3;-1] AND y ∈ [-3;-1] This gives us a solid rectangle with corners at (-3, -3), (-1, -3), (-1, -1), and (-3, -1). This entire rectangular area is included.
Combination 2: x ∈ [-3;-1] AND y ∈ (1;3] This gives us another solid rectangle. The x values are from -3 to -1 (inclusive), and the y values are from just above 1 up to 3 (inclusive). The corners are (-3, 1), (-1, 1), (-1, 3), and (-3, 3). Note that the bottom boundary at y=1 would be dashed since y cannot be exactly 1 in this part of the union.
Combination 3: x ∈ (1;2) AND y ∈ [-3;-1] Here, the x values are between 1 and 2 (exclusive), and the y values are between -3 and -1 (inclusive). This creates a rectangle where the left and right boundaries (x=1 and x=2) are dashed. The corners are (1, -3), (2, -3), (2, -1), and (1, -1). The top boundary at y=-1 is solid.
Combination 4: x ∈ (1;2) AND y ∈ (1;3] Finally, the x values are between 1 and 2 (exclusive), and the y values are between just above 1 and 3 (inclusive). This rectangle has dashed boundaries on all four sides: x=1, x=2, y=1, and y=3. The conceptual corners are (1, 1), (2, 1), (2, 3), and (1, 3).

So, for part b), you'll have four distinct rectangular regions. Two larger ones spanning x from -3 to -1, and two smaller ones spanning x from 1 to 2. These regions will be positioned relative to each other based on the y intervals. It looks like a few stacked blocks!

Part c) y ∈ (-∞; -1] ∪ [2; ∞)

Let's tackle the final part, c)! This one involves infinite intervals, which means we're dealing with regions that extend indefinitely on the graph. Again, our x condition is x∈[-3;-1] ∪ (1;2).

The y condition is y ∈ (-∞; -1] ∪ [2; ∞). This means y can be:

Any number less than or equal to -1 (extending downwards infinitely), OR
Any number greater than or equal to 2 (extending upwards infinitely).

We combine this with our x intervals. Similar to part b), we need to consider the intersections of the x and y conditions.

Region 1: x ∈ [-3;-1] AND y ∈ (-∞; -1] This defines a region that is a vertical strip bounded by x=-3 (solid line) and x=-1 (solid line). This strip extends downwards infinitely from y=-1 (solid line). So, it's like a rectangular strip that goes all the way down past the bottom of your graph paper.
Region 2: x ∈ [-3;-1] AND y ∈ [2; ∞) This is another vertical strip, bounded by x=-3 (solid line) and x=-1 (solid line). This strip extends upwards infinitely from y=2 (solid line). It goes all the way up off the top of your graph paper.
Region 3: x ∈ (1;2) AND y ∈ (-∞; -1] Here, the x values are between 1 and 2 (exclusive), so the vertical boundaries at x=1 and x=2 are dashed lines. This strip extends downwards infinitely from y=-1 (solid line).
Region 4: x ∈ (1;2) AND y ∈ [2; ∞) Finally, the x values are between 1 and 2 (exclusive), meaning x=1 and x=2 are dashed lines. This strip extends upwards infinitely from y=2 (solid line).

Graphically, this looks like two sets of vertical stripes. One set is between x=-3 and x=-1, extending infinitely both up and down, but only for y values below or equal to -1, and above or equal to 2. The second set is between x=1 and x=2 (with dashed boundaries), also extending infinitely up and down, and again, only for the specified y ranges. It covers a lot of area!

Putting It All Together: The Coordinate Plane

So, what does this all mean when you're actually drawing it? You'll need a standard Cartesian coordinate system with an x-axis and a y-axis. The key is to pay close attention to the intervals and whether the endpoints are included or excluded. Solid lines indicate that points on the boundary are part of the set, while dashed lines indicate that points on the boundary are not part of the set.

For the x condition x∈[-3;-1] ∪ (1;2), you'll draw vertical lines. You'll have solid vertical lines at x=-3 and x=-1. Then, you'll have dashed vertical lines at x=1 and x=2. The regions between these lines are where your points can exist.

Then, for each part (a, b, c), you overlay the y conditions. These translate into horizontal boundaries.

Solid horizontal lines mean the y value is included (like y=-1 in y ∈ (-∞; -1]).
Dashed horizontal lines mean the y value is excluded (like y=1 in y ∈ (1;3]).

Remember that union ∪ means you combine all the valid regions. If a point (x,y) satisfies the x condition AND any of the y conditions (for that part), it belongs to the set.

It’s like creating a blueprint for a set of points. You start with the allowed x ranges, then you stack or overlay the allowed y ranges, making sure to respect whether the boundaries are solid or dashed. The final picture is the collection of all points that satisfy both the x and y criteria simultaneously for each specific case (a, b, or c).

Don't be afraid to sketch it out! Using different colors for different boundaries or regions can also help keep things clear. The goal is to visually represent the solution set accurately. Practice makes perfect, guys, so try drawing these out yourself for different interval combinations. You'll get the hang of it in no time!