If g is an integer greater than 1 and ( g^3/(g + 1)^2 ) < 1, what is the possible value of g?
To find the possible value of g that satisfies the given inequality, let's analyze the inequality step by step:
The inequality is: g^3 / (g + 1)^2 < 1.
First, let's remove the denominator by multiplying both sides of the inequality by (g + 1)^2:
(g^3 / (g + 1)^2) * (g + 1)^2 < 1 * (g + 1)^2.
This simplifies to:
g^3 < (g + 1)^2.
Next, let's expand the right side of the inequality:
g^3 < g^2 + 2g + 1.
To solve this inequality, we can rearrange the terms to one side:
g^3 - g^2 - 2g - 1 < 0.
Now, since g is an integer greater than 1, we can test the values of g starting from 2 and onwards to find the values that satisfy the inequality. We can also use a graphing calculator or software to graph the equation and see where it is negative. Let's solve it:
For g = 2:
(2)^3 - (2)^2 - 2(2) - 1 = 8 - 4 - 4 - 1 = -1. This value does not satisfy the inequality.
For g = 3:
(3)^3 - (3)^2 - 2(3) - 1 = 27 - 9 - 6 - 1 = 11. This value satisfies the inequality.
For g = 4:
(4)^3 - (4)^2 - 2(4) - 1 = 64 - 16 - 8 - 1 = 39. This value also satisfies the inequality.
For g = 5:
(5)^3 - (5)^2 - 2(5) - 1 = 125 - 25 - 10 - 1 = 89. This value satisfies the inequality.
As we can see, the values of g that satisfy the inequality are 3, 4, and 5.
Therefore, the possible values of g are 3, 4, and 5.