Solve for x
x\in \left(0,7\right)
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-7x+x^{2}<0
Multiply the inequality by -1 to make the coefficient of the highest power in 7x-x^{2} positive. Since -1 is negative, the inequality direction is changed.
x\left(x-7\right)<0
Factor out x.
x>0 x-7<0
For the product to be negative, x and x-7 have to be of the opposite signs. Consider the case when x is positive and x-7 is negative.
x\in \left(0,7\right)
The solution satisfying both inequalities is x\in \left(0,7\right).
x-7>0 x<0
Consider the case when x-7 is positive and x is negative.
x\in \emptyset
This is false for any x.
x\in \left(0,7\right)
The final solution is the union of the obtained solutions.
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Limits
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