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$$\text{Find all positive values of c so that the inequality x^2-6x+c<0 has real solutions for x. Express your answer in interval notation. }$$

Jul 18, 2019

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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

Jul 18, 2019