Solve the inequality x + 3 < x^2 + 2x + 14. Give your answer in interval notation.
x + 3 < x^2 + 2x + 14 rearrange as
x^2 + x - 11 > 0
Let's solve this
x^2 + x - 11 = 0
x^2 + x = 11 complete the square on x
x^2 + x + 1/4= 11 + 1/4
(x + 1/2)^2 = 45/4 take both roots
x + 1/2 = sqrt (45) / 2 or x + 1/2 = -sqrt (45) / 2 { sqrt (45) = 3sqrt (5) }
x = [ 3 sqrt (5) - 1 ] / 2 or x = [ -3sqrt (5) - 1 ] / 2
These two answers are the x intercepts of a parabola that turns upward
Every real number less than the second answer will be in the solution and every real number greater than the first answer will be in the solutin ....so.....
x = ( -inf , [-3 sqrt 5 - 1 ] / 2 ) U ( [ 3sqrt (5) - 1 ] / 2 , inf )