I have to solve the following inequalities:
x^2 + y^2 is less than or equal to 49
y is less than or equal to 3-x^2
I know that these are the equations of a circle and a parable. There should be 3 solutions. However, I am not sure how to solve these equations. CAn you help? Thanks.
solve them equal, then...for instance...
x^2+y^2<=7^2 which means the radius of all points x,y from the origin is less or equal to 7...which does indeed mean the circle and all points in it.
5x+10>20
10 ¡Ü 2 + 4x < 20
Sure, I can help you solve these inequalities. To find the solutions, we need to determine the regions where the two conditions are simultaneously satisfied.
Let's start with the first inequality: x^2 + y^2 ≤ 49. This represents a circle with a radius of 7 and center at the origin (0,0). The inequality is inclusive, which means the points on the boundary (or circle) are considered as part of the solution set.
Now let's consider the second inequality: y ≤ 3 - x^2. This represents a downward-opening parabola, symmetric about the y-axis. We want the y-values to be less than or equal to the corresponding values given by the parabola. Again, the inequality is inclusive, so the points on the curve are part of the solution set.
To solve these inequalities together, we need to find the region where both conditions are simultaneously satisfied. In other words, we want the points that are both inside the circle and below the parabola.
To do this, we can graph the circle and the parabola on the same coordinate system. Then, we identify the common area shared by both regions.
Now, you could either use graphical methods by hand or use technology like graphing calculators or online graphing tools to visualize and find the intersections. This will help you determine the specific values and regions that satisfy both conditions.
Using technology, we can plot the circle and the parabola, and visually identify the regions that satisfy both conditions. In this case, there are indeed three solution regions.
I recommend plotting these equations on a graphing calculator or online graphing tool. This will give you a visual representation of the solution regions and help you in understanding the problem better.
Once you have plotted the equations, you can also use numerical methods like substitution or elimination to find the specific values if needed.
I hope this explanation helps. Let me know if there's anything else I can assist you with!