Which of the following values for n makes the inequality 2n−−√<5
true?(1 point)
Responses
8
8
12.5
12.5
18
18
16
16
To solve the inequality, we can start by isolating the square root term:
2n - √(n) < 5
Next, we can square both sides of the inequality:
(2n - √(n))^2 < 5^2
4n^2 - 4n√(n) + n < 25
4n^2 - 4n√(n) + n - 25 < 0
This is a quadratic inequality, which can be solved by factoring or using the quadratic formula. However, for simplicity, we can plug in the given values to check which ones satisfy the original inequality:
For n = 8:
2(8) - √(8) < 5
16 - 2.828 < 5
13.172 < 5 (This is false)
For n = 12.5:
2(12.5) - √(12.5) < 5
25 - 3.535 < 5
21.465 < 5 (This is false)
For n = 18:
2(18) - √(18) < 5
36 - 4.242 < 5
31.758 < 5 (This is false)
For n = 16:
2(16) - √(16) < 5
32 - 4 < 5
28 < 5 (This is false)
None of the given values for n make the inequality true.
What is the correct answer
Apologies for the confusion. I made an error in my previous response.
To solve the inequality 2n - √n < 5, we can isolate the square root term:
√n > 2n - 5
Now we can square both sides of the inequality:
n > (2n - 5)^2
n > 4n^2 - 20n + 25
Next, we can rearrange the equation and solve for n:
4n^2 - 20n + 25 - n < 0
4n^2 - 21n + 25 < 0
To factor this quadratic inequality, we can use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
For the quadratic equation 4n^2 - 21n + 25, we have a = 4, b = -21, and c = 25. Plugging these values into the quadratic formula:
n = (-(-21) ± √((-21)^2 - 4(4)(25))) / (2(4))
n = (21 ± √(441 - 400)) / 8
n = (21 ± √41) / 8
Since we're looking for solutions where n > 0, we only consider the positive root:
n = (21 + √41) / 8
n ≈ 3.09
Therefore, the correct answer is n ≈ 3.09.