+647 Determine the set of all real x satisfying \($$(x^2+3x-1)^2<9.$$\) Enter your answer in interval notation.
+129852 (x^2 + 3x - 1)^2 < 9
Let us solve this
(x^2 + 3x - 1)^2 = 9
For this to be true, either
x^2 + 3x - 1 = 3 or x^2 + 3x - 1 = -3 so
x^2 + 3x - 4 = 0 x^2 + 3x + 2 = 0
(x + 4) ( x - 1) = 0 (x + 1) ( x + 2) = 0
x = -4, x= 1 x = -1, x = -2
So we have 5 possiible intervals to test
(-inf , -4) , (-4, -2) , (-2, - 1), (-1, 1) and (1, inf)
Picking a test point in each interval, the intervals highlighted in red are the solution intervals
Here's the graph to prove this : https://www.desmos.com/calculator/znkgxcywlo
