Which number is a solution of the inequality?
12 < y (8 - y)
A. 0
B. 1
C. 2
D. 3
Does anyone have the correct answers for the whole solving inequalities unit test? it would be great if you could please help me with it!
Well, well, well, it seems you're in need of a little clown humor! While I can't give you the answers to your test, I can help you with the first question. Let's solve it together!
So, we have the inequality 12 < y (8 - y). To find the solution(s), we need to solve the quadratic inequality. Let's simplify it first:
12 < 8y - y^2
Rearranging to get the quadratic in standard form:
y^2 - 8y + 12 > 0
Now, let's factor this quadratic:
(y - 2)(y - 6) > 0
So, we have two factors, (y - 2) and (y - 6). To determine the signs and intervals where the inequality holds true, we need to consider the signs of these factors.
When (y - 2) > 0 and (y - 6) > 0, both factors are positive. This occurs when y > 6.
When (y - 2) < 0 and (y - 6) < 0, both factors are negative. This occurs when y < 2.
Now, let's consider when the signs of the factors are different.
When (y - 2) > 0 and (y - 6) < 0, the first factor is positive and the second is negative. This occurs when y > 6 or y < 2.
When (y - 2) < 0 and (y - 6) > 0, the first factor is negative and the second is positive. This does NOT satisfy the inequality.
So, the solution to the inequality is y > 6 or y < 2. Now, let's see which of the given numbers, 0, 1, 2, or 3, satisfy this solution. Can you figure it out?
To determine which number is a solution of the inequality 12 < y(8 - y), we can substitute each answer choice into the inequality and check if the inequality holds true.
A. 0: Substitute y = 0 into the inequality:
12 < 0(8-0)
12 < 0
This is not true, so 0 is not a solution.
B. 1: Substitute y = 1 into the inequality:
12 < 1(8-1)
12 < 1(7)
12 < 7
This is not true, so 1 is not a solution.
C. 2: Substitute y = 2 into the inequality:
12 < 2(8-2)
12 < 2(6)
12 < 12
This is not true, so 2 is not a solution.
D. 3: Substitute y = 3 into the inequality:
12 < 3(8-3)
12 < 3(5)
12 < 15
This is true, so 3 is a solution.
Therefore, the correct answer is D. 3.
To find a solution for the inequality 12 < y(8 - y), we can follow these steps:
Step 1: Expand the expression y(8 - y) using the distributive property.
12 < 8y - y²
Step 2: Rearrange the equation to have a quadratic expression on one side equal to zero.
0 < y² - 8y + 12
Step 3: Factorize the quadratic expression.
0 < (y - 2)(y - 6)
Step 4: Set each factor equal to zero and solve for y.
y - 2 = 0 -> y = 2
y - 6 = 0 -> y = 6
Step 5: Determine the intervals where the inequality is true.
Since we have "<" symbol, it means the inequality is true for the values of y that are between the two solutions we found, which are 2 and 6. In other words, 2 < y < 6.
Now, let's check each given option to see if any of them falls within the range of 2 < y < 6.
Option A: 0 is not within the range of 2 < y < 6.
Option B: 1 is not within the range of 2 < y < 6.
Option C: 2 is within the range of 2 < y < 6, so it is a solution.
Option D: 3 is within the range of 2 < y < 6, so it is a solution.
Based on the analysis above, options C and D are solutions to the inequality 12 < y(8 - y).
Regarding your second question about the whole solving inequalities unit test, as an AI language model, I don't have access to specific test answers. It's best to study your materials, understand the concepts, and attempt the questions using the methods taught in your course. If you're stuck on a specific question, I can certainly help explain the concepts or guide you through the steps to find the solution.