Which number is a solution to the inequality?
6 > z(10 - z)
a. 0
b. 1
c. 2
d. 3
To find the solution to the inequality, we need to solve the inequality for z.
Let's distribute the z to both terms inside the parentheses:
6 > 10z - z^2
This can be rearranged into the quadratic inequality form:
z^2 - 10z + 6 < 0
Now, let's factorize the quadratic equation:
(z - 2)(z - 3) < 0
To determine the solutions, we need to consider when the left side of the inequality is less than zero.
From the factored equation, we see that the inequality changes sign at z = 2 and z = 3. The inequality is negative in between these two values and positive outside these values.
Therefore, the solutions are 2 > z > 3.
The only integer that satisfies this condition is 3.
So, the number that is a solution to the inequality is d. 3.
To find which number is a solution to the inequality 6 > z(10 - z), we need to substitute each of the given options (0, 1, 2, 3) into the inequality and check if it holds true.
Let's substitute option (a), z = 0, into the inequality:
6 > 0(10 - 0)
6 > 0(10)
6 > 0
The inequality holds true for z = 0.
Now let's substitute option (b), z = 1, into the inequality:
6 > 1(10 - 1)
6 > 1(9)
6 > 9
The inequality does not hold true for z = 1.
Next, let's substitute option (c), z = 2, into the inequality:
6 > 2(10 - 2)
6 > 2(8)
6 > 16
The inequality does not hold true for z = 2.
Finally, let's substitute option (d), z = 3, into the inequality:
6 > 3(10 - 3)
6 > 3(7)
6 > 21
The inequality does not hold true for z = 3.
Therefore, the only number that is a solution to the inequality is z = 0. Thus, the answer is option (a).
To find the solution to the inequality 6 > z(10 - z), we can solve it step-by-step.
First, we can expand the right side of the inequality:
6 > 10z - z^2
Next, we can rearrange the equation to bring all terms to one side:
0 > z^2 - 10z + 6
Now, we can solve the quadratic equation by factoring or using the quadratic formula. In this case, the quadratic does not factor easily, so let's use the quadratic formula. The quadratic formula states that for an equation of 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 in these values, we get:
z = (-(-10) ± √((-10)^2 - 4(1)(6))) / (2(1))
z = (10 ± √(100 - 24)) / 2
z = (10 ± √76) / 2
z = (10 ± 2√19) / 2
z = 5 ± √19
So, the solutions to the inequality are z = 5 + √19 and z = 5 - √19.
Now, let's check which of the given options satisfy the inequality:
a. 0: 6 > 0(10 - 0) => 6 > 0, which is true
b. 1: 6 > 1(10 - 1) => 6 > 9, which is false
c. 2: 6 > 2(10 - 2) => 6 > 16, which is false
d. 3: 6 > 3(10 - 3) => 6 > 21, which is false
From the options given, the only solution to the inequality is z = 0, so the answer is option a.