+2653 Find all $x$ that satisfy the inequality $(2x+10)(x+3)<(3x+1)(x+6)$. Express your answer in interval notation.
+129895 Simplify as
2x^2 + 16x + 30 < 3x^2 + 19x + 6
Rearrange as
x^2 + 3x - 24 > 0 (1)
Express as an equality
x^2 + 3x - 24 = 0
x^2 + 3x + 9/4 = 24 + 9/4
(x + 3/2)^2 = 105/4 take both roots
x + 3/2 = sqrt (105) /2 x + 3/2 = -sqrt(105)/2
x = [ -3 + sqrt (105) ] / 2 ≈ 3.6 x = [ -3 -sqrt (105) ] / 2 ≈ -6.6
Our answer comes from either ( ≈ -6.6 , ≈ 3.6) or (-inf, ≈ -6.6) U ( ≈ 3.6 , inf)
Testing x =0 in (1) makes it false
So
The solution comes from (-inf, [-3-sqrt (105) ] / 2) U [ -3 + sqrt (105) ] / 2 , inf)
