6 > z (10 - z)
To simplify the given expression, let's first distribute the -z through the parentheses:
6 > 10z - z^2
Next, let's rearrange the terms to get the equation in standard quadratic form:
z^2 - 10z + 6 > 0
Now we have a quadratic inequality. We can solve this inequality by finding the roots of the equation:
(z - a)(z - b) > 0
First, let's find the roots of the equation z^2 - 10z + 6 = 0:
z = (-(-10) ± sqrt((-10)^2 - 4(1)(6))) / (2(1))
z = (10 ± sqrt(100 - 24)) / 2
z = (10 ± sqrt(76)) / 2
z = (10 ± 2sqrt(19)) / 2
z = 5 ± sqrt(19)
So the roots of the equation are 5 + sqrt(19) and 5 - sqrt(19).
Now, let's analyze the inequality by considering the sign of each factor:
When z < 5 - sqrt(19) or z > 5 + sqrt(19), the inequality is true because the product of two negative numbers is positive.
When 5 - sqrt(19) < z < 5 + sqrt(19), the inequality is false because the product of a positive number and a negative number is negative.
Therefore, the solution to the inequality 6 > z(10 - z) is:
z < 5 - sqrt(19) or z > 5 + sqrt(19)