Which number is a solution of the inequality?
6>z(10-z)
A)0
B)1
C)2
D)
6 > 10 z - z^2
well if z = 0, the right hand side is 0
and
6 is > 0
so A works
by the way, the others do not work ;)
Thank you!
You are welcome.
D)
To determine which number is a solution of the inequality 6 > z(10-z), we can use the process of elimination.
First, let's simplify the inequality:
6 > z(10-z)
Expanding the expression on the right side, we get:
6 > 10z - z^2
Rearranging the inequality by bringing all terms to one side, we have:
z^2 - 10z + 6 < 0
Now, we need to find the values of z that make this inequality true.
One way to solve this inequality is by factoring. However, let's use a different method called the quadratic formula.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -10, and c = 6. Plugging these values into the quadratic formula, we have:
z = (-(-10) ± √((-10)^2 - 4(1)(6))) / (2(1))
Simplifying further:
z = (10 ± √(100 - 24)) / 2
z = (10 ± √(76)) / 2
z = (10 ± √(4 * 19)) / 2
z = (10 ± 2√19) / 2
z = 5 ± √19
Now, looking at the answer choices, we can check which values satisfy the inequality.
A) 0: 6 > 0(10-0) -> 6 > 0 (True)
B) 1: 6 > 1(10-1) -> 6 > 9 (False)
C) 2: 6 > 2(10-2) -> 6 > 16 (False)
D) 5 + √19: 6 > (5+√19)(10-(5+√19)) -> 6 > (5+√19)(10-5-√19) -> 6 > (5-√19)(5+√19) -> 6 > (25-19) -> 6 > 6 (False)
By checking each answer choice, we can see that the only number that satisfies the inequality is A) 0.
Therefore, the solution to the inequality 6 > z(10-z) is z = 0.