+1207 Find all values of \(x\) that satisfy \(x(x+7.5) > 38.5\). Give your answer in interval notation.
+129852 x ( x + 7.5) > 38.5
x^2 + 7.5x > 38.5
x^2 + 7.5x - 38.5 > 0 multiply through by 2
2x^2 + 15x - 77 > 0 factor
(2x - 7) ( x + 11) > 0 set each factor to 0
2x - 7 = 0 x + 11 = 0
2x = 7 x = - 11
x = 7/2
We have three possible intervals that make the original inequality true
(-infinity, -11) or ( -11, 7/2) or (7/2, infinity)
Testing 0 in the middle interval makes the inequality false
So....the two intervals that make the original inequality true are
(-infinity, -11) U ( 7/2, infinity )
