+1678 Find all values of $x$ that satisfy
\[5x - 1 < (x + 1)^2 + 7x - 3.\]
+129845 Simplify this as
5x - 1 < x^2 + 2x + 1 + 7x - 3
5x - 1 < x^2 + 9x - 2
0 < x^2 - 4x - 1
So
x^2 - 4x - 1 > 0
This is a parabola that turns upward......using the Quadratic Formula, we can find where the parabola intersects the x axis......the solution will lie on (-inf , root 1) U (root 2, inf)
So
Root 1 = [ 4 - sqrt ( (-4)^2 - 4 (1) (-1) } / 2 = [ 4 - sqrt (20) ] / 2 = [4 - sqrt (4 * 5) ] / 2 =
[4 - 2sqrt (5) ] / 2 = 2 - sqrt 5
And by the conjugate property, Root 2 = 2 + sqrt 5
So.....the solution is ( inf, 2 -sqrt 5) U ( 2 + sqrt 5, inf)
