Question on properties of inequalities?

Question

(x^2-3x-3)^2>(x^2+7x-13)^2

Can you just take the sqaure root of both sides, or do you have to multiply it all out and then solve?

Steve A that means that 0 works and it doesnt, plug it in.

Answer 1

Take all terms to the R.H.S. and use the
a^2-b^2=(a-b)(a+b) formula to factorize and then solve the inequality

Answer 2

You can sqrt both sides, but this will result in:

|x²-3x-3| > |x²+7x-13|
which has to be broken up in little pieces...

LHS has roots (3±sqrt21)/2, or approx. -0.791 and 3.791
RHS has roots (-7±sqrt101)/2, or approx. -8.525 and 1.525.

So...

1) For x <= (-7-sqrt101)/2 = -8.525, both LHS and RHS have positive stuff between the "| |", so we have
x²-3x-3 > x²+7x-13
---> x<1 which is always true under the above restriction.
So (-infinity, (-7-sqrt101)/2] is a set of solutions.

2) for x between (-7-sqrt101)/2 = -8.525 and (3-sqrt21)/2 = -0.791, the RHS has negative stuff between the vertical lines, so
x²-3x-3 > -(x²+7x-13)
---> 2x² + 4x - 16 > 0
---> (2x-4)(x+4) >0
---> x>2 or x<-4 subject to the given restriction, -8.525<=x<=-0.791.
So [(-7-sqrt101)/2, -4) is another set of solutions.

3) If x is between -.791 and 1.525, then our equation becomes
-(x²-3x-3) > -(x²+7x-13)
---> x>1
yielding (1, (-7+sqrt101)/2] as another set of solutions.

4) If x is between 1.525 and 3.791, then
-(x²-3x-3) > x²+7x-13
---> -x² + 3x + 3 > x² + 7x - 13(-7±sqrt101)/2
---> 0 > 2x² + 4x - 16
---> 0 > (2x-4)(x+4)
---> -4 which gives [(-7+sqrt101)/2 , 2) as a set of solutions.

5) Finally, for x larger than any of the roots (x>3.791), we're back to the same case as (1) above which says x<1. Impossible under this constraint.

So, adding up these 5 cases, our solution set is
(-infinity, -4) U (1,2)

In retrospect, Keeper (below)'s solution is way, way easier... This results in
(-10x + 10)(2x² + 4x - 16) > 0
---> (1-x)(2x-4)(x+4)>0
which gets at the solution much faster.

Answer 3

In this case, you can take the square root of both sides as a first step.
x^2 -3x -3 > x^2 +7x -13
10 > 10x
1 > x
Because it's squared, check what happens at -1 also.
You find x > -1
So the final answer is: 1 > x > -1

Answer 4

I'd take the square root of each side as a first guess, but then I'd also multiply it out and solve. The reason is, if you take the square root you're losing information about +/- sqrt.

Answer 5

Go to youtube, type nutshellmath and search. Lots of math tutor videos...