Which number is a solution of the inequality? 6 > z(10-z)

Answer 1

To find which number is a solution of the inequality 6 > z(10-z), we can first simplify the inequality by expanding the expression on the right side:

6 > 10z - z^2

Next, rearrange the inequality to have a quadratic equation on the right side:

z^2 - 10z + 6 < 0

Now we can solve this quadratic inequality. Since the coefficient of the quadratic term is positive, the parabola opens upward and the expression is negative between its roots. To find the roots, we can factor the quadratic or use the quadratic formula:

(z - 2)(z - 3) < 0

The roots are z = 2 and z = 3. We can create a sign chart to determine when the expression is negative or positive:

- + – +

2 3

From the sign chart, we can see that the expression is negative between the roots z = 2 and z = 3.

Therefore, the values of z that satisfy the inequality 6 > z(10-z) are the numbers between 2 and 3, excluding 2 and 3.

Answer 2

Which number is a solution of the inequality? 6 > z(10-z)

1. 0
2. 1
3. 2
4. 3

Answer 3

To determine which number is a solution of the inequality 6 > z(10-z), we substitute each of the given numbers into the inequality.

1. 0: 6 > 0(10-0) simplifies to 6 > 0, which is true.
2. 1: 6 > 1(10-1) simplifies to 6 > 9, which is false.
3. 2: 6 > 2(10-2) simplifies to 6 > 16, which is false.
4. 3: 6 > 3(10-3) simplifies to 6 > 21, which is false.

Therefore, out of the given numbers, only 0 is a solution of the inequality.

Answer 4

Which number is a solution of the inequality? 6 > z(10-z)

Answer 5

To find which number is a solution of the inequality 6 > z(10-z), we can solve the inequality by setting it equal to zero and finding the values of z that make the inequality true.

First, let's set the inequality equal to zero:

z(10-z) - 6 > 0

Next, let's multiply out the expression on the left side:

10z - z^2 - 6 > 0

Rearranging the terms, we have:

-z^2 + 10z - 6 > 0

Now, let's factor the quadratic expression:

-(z^2 - 10z + 6) > 0

The quadratic expression does not factor nicely, so we can use the quadratic formula:

z = (-b ± sqrt(b^2 - 4ac)) / (2a)

For the quadratic expression z^2 - 10z + 6, the coefficients are a = -1, b = 10, and c = 6:

z = (-10 ± sqrt(10^2 - 4(-1)(6))) / (2(-1))

Simplifying further:

z = (-10 ± sqrt(100 + 24)) / (-2)

z = (-10 ± sqrt(124)) / (-2)

z = (-10 ± sqrt(4 * 31)) / (-2)

z = (-10 ± 2sqrt(31)) / (-2)

Simplifying again:

z = 5 ± sqrt(31)

Since we are looking for values of z that make the inequality true, we need to find where the expression is greater than zero. This occurs when:

z < 5 - sqrt(31) or z > 5 + sqrt(31)

Therefore, the numbers 5 - sqrt(31) and 5 + sqrt(31) are solutions to the inequality.