graph the system of inequalities: (
x-3)^2/9 + (y+2)^2/4<=1
and (x-3)^2+(y+2)^2>=4
I do not know how to simplify this down enough to graph it
The first appears to be a all within a circle of radius 1, centered at (3,-2). Is that correct?
The second is a outside a circle of radius 2, centered at 3,-2
You do not need to simplify it, you need to RECOGNIZE a conic equation when you see one. Circle, parabola, hyperbola, and ellipse equations you should recognize those on sight. If you do not, get the standard forms on flashcards (they work) and MEMORIZE them.
I thought the first was an ellipse and the second was a circle
Go on: wolframalpha dot com
When page be open in rectangle type:
( x-3)^2/9 + (y+2)^2/4<=1 , (x-3)^2+(y+2)^2>=4
and click option =
After few seconds you will see everything about your inequalities,including graph.
Thank you both so much.
To graph the system of inequalities, we need to simplify the equations and identify the shapes involved. Let's start with each inequality separately:
1. (x-3)^2/9 + (y+2)^2/4 <= 1
This is an equation of an ellipse. To graph it, we can follow these steps:
Step 1: Find the center of the ellipse. In this case, the center is (3, -2), which is the opposite sign of what is inside the parentheses.
Step 2: Determine the lengths of the major and minor axes. For the x-term, the square root of the denominator gives us 3, which represents the length of the semi-major axis. For the y-term, the square root of the denominator gives us 2, which represents the length of the semi-minor axis.
Step 3: Sketch the ellipse centered at (3, -2), with the major and minor axes' lengths as determined in Step 2.
2. (x-3)^2 + (y+2)^2 >= 4
This is an equation of a circle. To graph it, we can follow these steps:
Step 1: Find the center of the circle. In this case, the center is (3, -2).
Step 2: Determine the radius of the circle. The square root of 4 gives us 2, which represents the radius.
Step 3: Sketch the circle centered at (3, -2) with a radius of 2.
Now, we have the ellipse and the circle graphed separately. To find the graph of the system of inequalities, we need to consider the overlapping region.
The overlapping region is formed where the ellipse and the circle intersect or coincide. Shade the region where both inequalities are satisfied, which means the region where the graph of the ellipse is inside or on the boundary of the circle.
Remember to also include any boundaries as dashed lines or solid lines, depending on whether the inequality includes an equal sign or not.
For a more accurate graph, you can plot additional points around the ellipse and the circle to determine the precise shape of the overlapping region.
I hope this helps you graph the system of inequalities!