solve the inequality, solution in interval notation.

Question

x^2(x-6)>0   and (x-4)^2 / (x-6)(x+6) >0  These are two different problems - thank you

Answer 1

a) Since a perfect square is always positive, x-6>0 so x must be greater than 6: x>6.

b) The inequality requires x not to be equal to 6 or -6. The numerator is always positive because it's a perfect square. This means that the denominator determines the sign. When x-6<0, x<6 so x>6 is not solution. When both factors of the denominator are positive the result is positive; when both are negative the result is positive. When both are positive x>6 and -6, so x>6 takes prominence. When both are negative x-6<0 and x+6<0, so x<-6 is prominent. Therefore the solution is x>6 or x<-6. Another way of expressing the solution is that x must not be between -6 and 6 inclusive.