For what real values of x is -4 < x^4 + 4x^2 < 21 + 6x^2 satisfied? Express your answer in interval notation.
+128475 We have two inequalities
x^4 + 4x^2 > -4 also x^4 + 4x^2 < 6x^2 + 21
x^4 + 4x^2 + 4 > 0 x^4 - 2x^2 - 21 < 0 complete the square on x
(x^2 + 2)^2 > 0 x^4 - 2x^2 + 1 - 21 - 1 < 0
x^2 + 2 is > 0 for all x (x^2 - 1)^2 - 22 < 0
(x^2 - 1)^2 < 22
x^2 - 1 < ±sqrt (22)
So
x^2 - 1 < sqrt (22) and x^2 - 1 < -sqrt (22)
x^2 < sqrt (22) + 1 x^2 < 1 -sqrt 22
x < sqrt [ sqrt (22) + 1 ] x < sqrt [ 1 -sqrt (22 ] not a real number
x > - sqrt [ sqrt (22) + 1] x > - sqrt [ 1 -sqrt (22) ] not a real number
The most restrictive interval is
x = ( -sqrt [ sqrt(22) + 1 ] , sqrt [ sqrt (22) + 1 ] )
