7x-x2>0
To solve the inequality \(7x - x^2 > 0\), we can first rearrange it:
Rewrite the inequality as: \[ -x^2 + 7x > 0 \]
Factor out a negative sign (which reverses the inequality): \[ x^2 - 7x < 0 \]
Now, factor the quadratic: \[ x(x - 7) < 0 \]
The critical points are \(x = 0\) and \(x = 7\). We can use these critical points to determine the intervals to test.
The intervals to test are:
- \( (-\infty, 0) \)
- \( (0, 7) \)
- \( (7, \infty) \)
Now we test each interval:
For \(x < 0\) (e.g., \(x = -1\)): \[ (-1)(-1 - 7) = -1 \times -8 = 8 > 0 \quad \text{(not part of the solution)} \]
For \(0 < x < 7\) (e.g., \(x = 1\)): \[ (1)(1 - 7) = 1 \times -6 = -6 < 0 \quad \text{(part of the solution)} \]
For \(x > 7\) (e.g., \(x = 8\)): \[ (8)(8 - 7) = 8 \times 1 = 8 > 0 \quad \text{(not part of the solution)} \]
The solution is where the product is less than zero, which occurs in the interval \( (0, 7) \).
Thus, the solution to the inequality \(7x - x^2 > 0\) is: \[ \boxed{(0, 7)} \]
