\(\text{Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. }\)
+128408 x^2 - 6x + c < 0
Set this = 0
If this has real solutions, then the discriminant is ≥ 0
So
6^2 - 4c ≥ 0
36 - 4c ≥ 0
36 ≥ 4c
c ≤ 9
When c = 9...the parabola will have its vertex at (3,0)......so....since it turns upward, this is the low point on the graph....so...it is not less than 0 at any point
So.....this implies that the inequality will be < 0 [ i.e., have real solutions for x ] when c = (0, 9)
Here's a graph to show this : https://www.desmos.com/calculator/frqkypfzn8
