Solving System Of Inequalities: A Coordinate Plane Approach
Hey guys! Today, we're diving into a fascinating problem in algebra: finding the set of points on a coordinate plane that satisfy a system of inequalities. Specifically, we'll be tackling the system x² + y² ≤ 16 and x + y ≥ 2. This involves understanding how inequalities translate into regions on the coordinate plane and how to identify the overlapping region that satisfies both. Buckle up, because we're about to embark on a visual and algebraic journey!
Understanding the Inequalities
Okay, let's break down these inequalities one at a time. It's super important to get a solid grasp of what each one represents graphically. This is the key to visualizing our solution set.
Decoding x² + y² ≤ 16
The first inequality, x² + y² ≤ 16, should ring a bell for anyone familiar with circles. Remember the standard equation of a circle centered at the origin (0, 0)? It's x² + y² = r², where 'r' is the radius. In our case, we have x² + y² ≤ 16, which looks a lot like a circle equation, doesn't it? If we rewrite 16 as 4², we can see that this inequality represents all points (x, y) whose distance from the origin is less than or equal to 4. Think of it as the circle itself (x² + y² = 16) and everything inside that circle.
So, what does this mean visually? Imagine drawing a circle on the coordinate plane with its center smack-dab at the origin and a radius of 4 units. The inequality x² + y² ≤ 16 represents all the points that lie either on this circle or inside it. It's like shading in the entire circular region, including the boundary.
The main keyword here is circle. We're dealing with a circular region defined by a radius of 4, and the inequality tells us we're interested in the points within and on the boundary of this circle. This understanding is crucial for visualizing the solution set later on.
Unpacking x + y ≥ 2
Now, let's tackle the second inequality: x + y ≥ 2. This one might look a bit different, but it's still quite manageable. This is a linear inequality, which means it represents a region bounded by a straight line. To understand which region, we first consider the line x + y = 2. This is the boundary line of our region.
How do we graph this line? We can rewrite the equation in slope-intercept form (y = mx + b) to make it easier to plot. Subtracting 'x' from both sides, we get y = -x + 2. This tells us the line has a slope of -1 and a y-intercept of 2. We can plot the y-intercept (0, 2) and then use the slope to find another point (e.g., move 1 unit to the right and 1 unit down to get the point (1, 1)). Draw a line through these points, and you've got the graph of x + y = 2.
But we're not just interested in the line itself; we're interested in the inequality x + y ≥ 2. This means we want all the points (x, y) that, when plugged into the left side, result in a value greater than or equal to 2. Which side of the line does this represent? We can test a point, like (0, 0), which is often the easiest. If we plug in (0, 0), we get 0 + 0 ≥ 2, which is false. Since (0, 0) does not satisfy the inequality, the region we want is the one opposite to (0, 0). This is the region above and to the left of the line x + y = 2.
The key keywords here are linear inequality and half-plane. The inequality defines a half-plane, which is a region bounded by a line. We need to determine which half-plane satisfies the inequality by either visualizing it or testing a point.
Finding the Solution Set: The Intersection
Alright, we've successfully deciphered what each inequality represents individually. Now comes the fun part: combining them to find the solution set! Remember, the solution set is the set of all points (x, y) that satisfy both inequalities simultaneously.
The Graphical Approach
The best way to visualize this is to graph both inequalities on the same coordinate plane. We've already discussed how to graph them: the circle x² + y² ≤ 16 and the half-plane x + y ≥ 2. The solution set is the region where these two shaded areas overlap. This overlapping region is the set of points that satisfy both inequalities.
Imagine shading the circle (including its interior) and then shading the half-plane above the line x + y = 2. The region where the shading is darkest—where the two shadings overlap—is your solution set. It's a segment of the circle, cut off by the line.
Describing the Region
The solution set is a circular segment. It's the portion of the circle x² + y² ≤ 16 that lies above the line x + y = 2. This region includes all the points inside the circle that are also on or above the line. Visually, it looks like a curved slice of a pie.
Why is this the solution? Because any point within this segment satisfies both conditions: it's within the circle (satisfying x² + y² ≤ 16) and it's on or above the line (satisfying x + y ≥ 2).
The key keyword here is intersection. The solution set is the intersection of the two regions defined by the individual inequalities. This concept is fundamental to solving systems of inequalities.
Algebraic Refinement (Optional)
While the graphical approach gives us a clear visual understanding of the solution, we can also explore the solution set algebraically. This can be a bit more challenging, but it provides a deeper insight.
Finding Intersection Points
To get a more precise description of the solution set, we can find the points where the circle and the line intersect. These points are the “corners” of our circular segment. To find them, we need to solve the system of equations:
- x² + y² = 16
- x + y = 2
We can solve this system using substitution or elimination. Let's use substitution. From the second equation, we can write y = 2 - x. Substituting this into the first equation, we get:
x² + (2 - x)² = 16
Expanding and simplifying, we get:
x² + 4 - 4x + x² = 16
2x² - 4x - 12 = 0
Dividing by 2, we have:
x² - 2x - 6 = 0
Now we can use the quadratic formula to solve for x:
x = [2 ± √(2² - 4(1)(-6))] / 2
x = [2 ± √(28)] / 2
x = [2 ± 2√7] / 2
x = 1 ± √7
So we have two x-values: x₁ = 1 + √7 and x₂ = 1 - √7. We can plug these back into the equation y = 2 - x to find the corresponding y-values:
y₁ = 2 - (1 + √7) = 1 - √7
y₂ = 2 - (1 - √7) = 1 + √7
Therefore, the intersection points are (1 + √7, 1 - √7) and (1 - √7, 1 + √7). These points mark the boundaries of our circular segment.
Describing the Solution Algebraically
While we can't write a simple algebraic expression for the entire solution set, finding the intersection points helps us understand its boundaries. We know the solution consists of all points (x, y) that satisfy both inequalities, and we now have the points where the boundary curves meet.
This part is a bit more advanced, and the graphical solution usually suffices for most purposes. But it’s cool to see how algebra can complement our visual understanding!
Key Takeaways
Phew! We've covered a lot of ground here, guys. Let's recap the key concepts we've learned:
- Inequalities and Regions: Inequalities define regions on the coordinate plane. Linear inequalities define half-planes, while inequalities involving x² and y² can define circles or other conic sections.
- Graphical Solutions: The best way to solve a system of inequalities is often graphically. Shade the regions defined by each inequality, and the solution set is the overlapping region.
- Intersection: The solution set of a system of inequalities is the intersection of the solution sets of the individual inequalities.
- Algebraic Refinement: While graphical solutions are often sufficient, algebraic methods can help refine our understanding, especially by finding intersection points.
Conclusion
Solving systems of inequalities can seem daunting at first, but by breaking it down step by step and visualizing the regions, it becomes much more manageable. The combination of graphical and algebraic approaches gives us a powerful toolkit for tackling these problems. Keep practicing, and you'll become a pro at navigating the coordinate plane!
I hope this explanation was helpful, guys! If you have any questions or want to explore more examples, feel free to ask. Happy solving!
