Which number is a solution of the inequality?
6 > z(10-z)
This inequality can be rewritten as z^2 - 10z + 6 > 0.
To solve this quadratic inequality, we can first find the zeros of the related quadratic equation, which is z^2 - 10z + 6 = 0.
By factoring or using the quadratic formula, the zeros are z ≈ 9.34 and z ≈ 0.66.
Next, we can plot these zeros on a number line to determine the solution intervals.
On the number line, we have:
0.66 9.34
Now we can test a value in each interval to determine if it satisfies the inequality.
If we test a value less than 0.66, such as z = 0, we get:
6 > 0(10-0)
6 > 0 (which is true)
If we test a value greater than 9.34, such as z = 10, we get:
6 > 10(10-10)
6 > 0 (which is false)
Thus, the solution to the inequality is z < 0.66 or z > 9.34.
To find the number that is a solution of the inequality 6 > z(10-z), we need to solve the quadratic inequality.
Step 1: Rearrange the inequality to have zero on one side:
z(10-z) - 6 > 0
Step 2: Expand the expression:
10z - z^2 - 6 > 0
Step 3: Reorder the terms:
-z^2 + 10z - 6 > 0
Step 4: Solve the quadratic equation by factoring or using the quadratic formula. In this case, let's use factoring:
(-z + 2)(z - 3) > 0
Step 5: Set each factor equal to zero and solve for z:
-z + 2 = 0 or z - 3 = 0
Solving these equations gives:
z = 2 or z = 3
Step 6: Construct a sign chart to determine the intervals where the inequality is true or false:
(-infinity, 2) (2, 3) (3, +infinity)
+ - +
Step 7: Determine the sign of the inequality in each interval:
- For z < 2, the expression (-z + 2)(z - 3) is positive, so it satisfies the inequality.
- For 2 < z < 3, the expression (-z + 2)(z - 3) is negative, so it does not satisfy the inequality.
- For z > 3, the expression (-z + 2)(z - 3) is positive, so it satisfies the inequality.
Step 8: Determine the solution:
From the sign chart, we can see that the inequality 6 > z(10-z) is satisfied when z < 2 or z > 3. Therefore, the numbers that are solutions to the inequality are z < 2 and z > 3.
To find a solution to the inequality 6 > z(10-z), we can solve it by breaking it into two parts:
1. Firstly, let's consider the inequality z(10-z) > 0.
To find the solution to this inequality, we need to identify the values of z that make the expression z(10-z) greater than zero.
We can do this by examining the sign of the expression for different ranges of z.
a) When z < 0, both z and (10-z) are negative, so the product z(10-z) is positive.
b) When 0 < z < 10, z is positive and (10-z) is positive, so the product is positive.
c) When z > 10, both z and (10-z) are positive, so the product z(10-z) is positive.
Hence, the inequality z(10-z) > 0 is satisfied for z < 0 or 0 < z < 10, or z > 10.
2. Now, let's look at the original inequality 6 > z(10-z).
We need to find the values of z that make the left side of the inequality greater than the right side.
a) If we consider z < 0, the right side of the inequality will be positive, but the left side (6) is greater. So, this range of z does not satisfy the inequality.
b) If we consider 0 < z < 10, the right side of the inequality is positive, but the left side (6) is still greater. So, this range of z does not satisfy the inequality.
c) Finally, if we consider z > 10, the right side of the inequality becomes negative. As the left side is positive (6), this range of z satisfies the inequality.
Therefore, the only value of z that is a solution to the inequality 6 > z(10-z) is z > 10.