Which number is a solution of the inequality? explain
6>z(10-z)
(1 point)
Responses
0
1
2
3
To find which number is a solution of the inequality 6 > z(10-z), we can substitute the given numbers from the choices into the inequality and see which ones satisfy the inequality.
Let's test each option:
- If we substitute 0 into the inequality, we get: 6 > 0(10-0) which simplifies to 6 > 0. This is true.
- If we substitute 1 into the inequality, we get: 6 > 1(10-1) which simplifies to 6 > 9. This is false.
- If we substitute 2 into the inequality, we get: 6 > 2(10-2) which simplifies to 6 > 16. This is false.
- If we substitute 3 into the inequality, we get: 6 > 3(10-3) which simplifies to 6 > 21. This is false.
Therefore, the only number that is a solution of the inequality is 0.
To find out which number is a solution to the inequality 6 > z(10-z), we can solve the inequality step by step.
Step 1: Expand and simplify the inequality.
6 > 10z - z^2
Step 2: Rearrange the inequality in descending order.
-z^2 + 10z - 6 > 0
Step 3: Factor the quadratic expression.
-(z - 2)(z - 3) > 0
Step 4: Determine the intervals where the expression is greater than zero (positive).
To do this, we make a sign chart by setting each factor equal to zero and determining the sign of the expression in each interval.
(-∞, 2) (2, 3) (3, +∞)
- + -
In the interval (-∞, 2), both factors are negative, resulting in a positive product (- * - > 0).
In the interval (2, 3), one factor is negative and the other is positive, resulting in a negative product (- * + < 0).
In the interval (3, +∞), both factors are positive, resulting in a positive product (+ * + > 0).
Step 5: Determine the valid solutions.
Since the question asks for a number that satisfies the inequality, we need to find the intervals where the expression is positive (+) and choose a value within those intervals.
From the sign chart, we can see that the only interval where the expression is positive (+) is (3, +∞). Therefore, the valid solutions are any number greater than 3.
Based on the options provided (0, 1, 2, and 3), none of them are valid solutions to the inequality.
To find the solution(s) of the given inequality 6>z(10-z), we need to solve for z.
First, let's distribute z to both terms inside the parentheses:
6 > 10z - z^2
Now, rearrange the terms to obtain a quadratic inequality in standard form:
z^2 - 10z + 6 < 0
To solve this inequality, we can factor or use the quadratic formula. However, since the inequality involves a strict less than sign (<) instead of less than or equal to (≤), we can use the fact that a quadratic inequality is negative between its roots.
To find the roots, set the equation equal to zero:
z^2 - 10z + 6 = 0
This quadratic equation doesn't factor nicely, so we'll use the quadratic formula:
z = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -10, and c = 6. Plugging these values into the quadratic formula gives us:
z = (10 ± √((-10)^2 - 4(1)(6))) / (2(1))
z = (10 ± √(100 - 24)) / 2
z = (10 ± √76) / 2
z = (10 ± √(4*19)) / 2
z = (10 ± 2√19) / 2
z = 5 ± √19
Since the inequality is less than zero, we want to find the values of z that make the expression on the left side negative. Using the fact that a quadratic is negative between its roots, we can determine the solution:
5 - √19 < z < 5 + √19
Thus, the solution to the inequality 6 > z(10-z) is:
5 - √19 < z < 5 + √19
None of the answer choices (0, 1, 2, 3) fall within the interval 5 - √19 < z < 5 + √19, so none of them satisfy the inequality.