Simplifying Inequalities

In summary, the given inequality, 1296(b^3) - 324(b^2) - 1008b + 108 > 0, is a third order polynomial with either one or three real roots. After finding the approximate values of these roots (-0.821, 0.105, 0.966), it can be concluded that the inequality is true for values of b between the first two roots and for b higher than the highest root.
  • #1
lokisapocalypse
32
0
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?
 
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  • #2
lokisapocalypse said:
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
 
  • #3


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
 

1. What is the purpose of simplifying inequalities?

Simplifying inequalities allows for easier understanding and comparison of values in mathematical expressions. It also helps to identify any patterns or relationships between the variables involved.

2. How do you simplify an inequality?

To simplify an inequality, you can follow the same rules as simplifying equations, such as combining like terms and using inverse operations. However, when multiplying or dividing by a negative number, the inequality sign must be flipped to maintain the correct direction of the inequality.

3. Can you solve an inequality like an equation?

No, inequalities cannot be solved like equations. Instead of finding the exact value of the variable, the goal is to find a range of values that satisfy the inequality. This is represented by a solution set or interval notation.

4. What are the key differences between solving equations and simplifying inequalities?

The key differences between solving equations and simplifying inequalities include the use of inequality signs instead of equal signs, the need to flip the inequality sign when multiplying or dividing by a negative number, and the use of solution sets or interval notation instead of a single value as the solution.

5. Why is it important to check your answer when simplifying inequalities?

It is important to check your answer when simplifying inequalities because flipping the inequality sign can lead to a different solution set than what was initially found. Additionally, it is always good practice to check for any potential errors in the simplification process.

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