x^2 > y^2 (1 Viewer)

[stressedessert04]

how to sketch this region? i took 4 cases but it ended up super convoluted anyone has a smart efficient way of solving it?

 

[ExtremelyBoredUser]

how to sketch this region? i took 4 cases but it ended up super convoluted anyone has a smart efficient way of solving it?

x^2 > y^2
x^2 - y^2 >0
(x-y)(x+y) >0

consider this normally;
x^2 = y^2
Square rooting this
+- x = +- y

This would be the same as an absolute value graph which was also flipped about the x axis. You can consider this as a piecewise function as well, y=-x and y = x or x = -y, x = y from the signs.

Now for this to be greater than 0, the values bounded, the values can not be above y = x or y=-x since they produce values that do not fit the equality. I really doubt you can solve this question through pure algebra, you have to have a graphical intuition of the operators then be able to understand what the inequality is asking for.

 

[CM_Tutor]

You know the boundaries are at and are outside the region (so dotted).

If then you have , which has no solution, so no part of the -axis is in the region.

If then you have , which has as solution any real , so all of the -axis except the origin is in the region.

So, you know which of the four regions are included or excluded and can sketch the region as shown above.

 

[stressedessert04]

x^2 > y^2
x^2 - y^2 >0
(x-y)(x+y) >0

consider this normally;
x^2 = y^2
Square rooting this
+- x = +- y

This would be the same as an absolute value graph which was also flipped about the x axis. You can consider this as a piecewise function as well, y=-x and y = x or x = -y, x = y from the signs.

Now for this to be greater than 0, the values bounded, the values can not be above y = x or y=-x since they produce values that do not fit the equality. I really doubt you can solve this question through pure algebra, you have to have a graphical intuition of the operators then be able to understand what the inequality is asking for.

View attachment 32887

oooh a lot of intuition involved but most of it clicked. thanks!

 

[CM_Tutor]

oooh a lot of intuition involved but most of it clicked. thanks!

You could also just test four points, one in each region, such as .