What Values of b Satisfy the Inequality 1296(b^3) - 324(b^2) - 1008b + 108 > 0?

[lokisapocalypse]

I have an inequality:

1296(b^3) - 324(b^2) - 1008b + 108 > 0.

I want to know for what values of b this inequality is true. Any suggestions?

 

[ehild]

I have an inequality:

1296(b^3) - 324(b^2) - 1008b + 108 > 0.

I want to know for what values of b this inequality is true. Any suggestions?

I do not think you can simplify the expression too mutch except factoring out 36.

Anyway, it is a third order polynomial, which goes from -infinity to + infinity, and having either one or three real roots. You can find them approximately(-0.821, 0.105, 0.966). As the polynomial is negative for very low values of b and positive for very high ones, it is positive between the first two roots and also for b higher than the highest root.

ehild

 

[wais]

To solve this inequality, we can start by factoring out a common factor of 108:

108(12b^3 - 3b^2 - 9b + 1) > 0

Next, we can factor the expression inside the parentheses using the grouping method:

108[(12b^3 - 3b^2) + (-9b + 1)] > 0

108[3b^2(4b - 1) - 1(9b - 1)] > 0

108(3b^2 - 1)(4b - 1) > 0

Now, we can find the critical values of b by setting each factor equal to 0:

3b^2 - 1 = 0 and 4b - 1 = 0

Solving for b, we get b = ±1/√3 and b = 1/4.

These values divide the number line into four intervals:

1) b < -1/√3
2) -1/√3 < b < 1/4
3) 1/4 < b < 1/√3
4) b > 1/√3

We can now test a value from each interval to see if it satisfies the inequality:

1) Let b = -2. Plugging this into the original inequality, we get:
1296(-2)^3 - 324(-2)^2 - 1008(-2) + 108 = -6912 + 1296 + 2016 + 108 = -3492 < 0
Since this value does not satisfy the inequality, we can eliminate this interval.

2) Let b = 0. Plugging this into the original inequality, we get:
1296(0)^3 - 324(0)^2 - 1008(0) + 108 = 108 > 0
Therefore, this interval is a solution to the inequality.

3) Let b = 1/3. Plugging this into the original inequality, we get:
1296(1/3)^3 - 324(1/3)^2 - 1008(1/3) + 108 = 144 - 36 - 336 + 108 = -120 < 0
Since this value does not satisfy the inequality, we can eliminate this interval