Which Values of x and y Satisfy the Inequality x^2y+y^2x > 6?

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The discussion focuses on finding values of x and y that satisfy the inequality x^2y + y^2x > 6. A suggested approach involves analyzing the related equality x^2y + y^2x = 6, which can be treated as a quadratic equation in y. By applying the quadratic formula, the boundary conditions can be determined, and graphing these can help visualize the regions where the inequality holds. It is noted that while graphical methods are useful, they are not the only means to understand the inequality, as the boundary provides precise regions in the x,y plane. Ultimately, the inequality simplifies for x > 0 into two distinct regions based on the derived quadratic expressions.
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For what values of x and , y is the following inequality satisfied:


x^2y+y^2x >6

I tried to give a proof and i went as far:

xy(x+y)>6 then what?
 
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evagelos said:
For what values of x and , y is the following inequality satisfied:


x^2y+y^2x >6

I tried to give a proof and i went as far:

xy(x+y)>6 then what?
The best way to handle complicated inequalities is to look first at the associated equality. Here, we start by looking at x^2y+ y^2x= 6[/itex]. You can think of that as a quadratic equation in y and solve using the quadratic formula: <br /> y= \frac{-x^2\pm\sqrt{x^4+ 24x}}{2x}<br /> <br /> Graph that on, say, a graphing calculator and it shows the boundary between &quot;&gt; 6&quot; and &quot;&lt; 6&quot;. Putting in one (x, y) point for each region will tell you which regions are &quot;&gt; 6&quot;.
 


HallsofIvy said:
The best way to handle complicated inequalities is to look first at the associated equality. Here, we start by looking at x^2y+ y^2x= 6[/itex]. You can think of that as a quadratic equation in y and solve using the quadratic formula: <br /> y= \frac{-x^2\pm\sqrt{x^4+ 24x}}{2x}<br /> <br /> Graph that on, say, a graphing calculator and it shows the boundary between &quot;&gt; 6&quot; and &quot;&lt; 6&quot;. Putting in one (x, y) point for each region will tell you which regions are &quot;&gt; 6&quot;.
<br /> <br /> You mean then, that there is no solid proof for the inequality but only graphic procedures?<br /> <br /> How about the inequalityx^3y+yx^3&amp;gt;10 can we use graphic procedures??
 


evagelos said:
You mean then, that there is no solid proof for the inequality but only graphic procedures?

No that's not what Halls said. He found the boundary which precisely identifies the region of the x,y plane you are looking for. Graphing it is optional (though very helpful in my opinion).

For x>0 it reduces to the union of :

<br /> y &gt; \frac{-x^2 + \sqrt{x^4+ 24x}}{2x}<br />

and

<br /> y &lt; \frac{-x^2 - \sqrt{x^4+ 24x}}{2x}<br />
 
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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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