Which number is a solution of inequality
10<y(7-y)
A. 0
B. 1
C. 2
D. 3
Well, solving inequalities can sometimes be a little "inequal" for me, but let's give it a go!
To find the solutions, we can start by expanding the expression:
10 < 7y - y^2
Now, let's rearrange it to form a quadratic equation:
y^2 - 7y + 10 > 0
Next, we can factorize the quadratic equation:
(y - 2)(y - 5) > 0
So, the inequality is satisfied when either:
y > 5 or y < 2
Looking at the choices, none of them fit that criteria. It seems like we don't have a number that satisfies this inequality among the options provided.
So, the answer is E. None of the above. Keep searching, my friend!
To find the solutions for the inequality 10 < y(7-y), we can first simplify the inequality.
10 < y(7-y)
10 < 7y - y^2
Rearranging the terms:
y^2 - 7y + 10 < 0
Now, we can factor the quadratic equation:
(y - 5)(y - 2) < 0
To find the solutions, we need to determine the values of y that make the expression (y - 5)(y - 2) less than zero.
The key concept here is the product of two numbers being less than zero. In order for the product to be negative, one factor must be positive and the other factor must be negative.
Considering the options given:
A. 0: (0 - 5)(0 - 2) = (-5)(-2) = 10 > 0 (not a solution)
B. 1: (1 - 5)(1 - 2) = (-4)(-1) = 4 > 0 (not a solution)
C. 2: (2 - 5)(2 - 2) = (-3)(0) = 0 (not a solution)
D. 3: (3 - 5)(3 - 2) = (-2)(1) = -2 < 0 (a solution)
Therefore, the number 3 is a solution to the inequality 10 < y(7-y). The correct choice is option D.
To find the solution to the inequality 10 < y(7-y), we can follow these steps:
Step 1: Simplify the expression on the right side of the inequality:
y(7-y) = 7y - y^2
Step 2: Rewrite the inequality:
10 < 7y - y^2
Step 3: Rearrange the inequality to bring all terms to one side in descending order:
y^2 - 7y + 10 > 0
Step 4: Factorize the quadratic expression:
(y - 5)(y - 2) > 0
Step 5: Determine the sign of each factor:
For (y - 5) > 0, y > 5
For (y - 2) > 0, y > 2
Step 6: Determine the positive intervals using a number line:
- ∞ 2] 5] ∞
Step 7: Determine the numbers in the positive intervals:
Numbers greater than 5 and numbers between 2 and 5.
From the given answer choices, the number that is a solution to the inequality is C. 2.
well, have you tried the choices to see which works?
for example, if y=0, then you have
10 < 0 (7-0)
10 < 0
This one fails. So try the others.