Proving Inequality: (6x^2 - 7sinx)/(3x+1) >1000

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In summary, the conversation discusses a question and solution related to proving an inequality for a specific equation. The speaker makes approximations and simplifies the equation to reach the conclusion that the inequality holds for x > 1008. The conversation ends with a suggestion for the listener to continue thinking for themselves.
  • #1
nhrock3
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i have this question and solution
2rdb1ae.jpg

i have a similar question which i can't solve similarly

i need to prove
(6x^2 - 7sinx)/(3x+1) >1000 for x>m and x>0
(6x^2 - 7sinx)/(3x+1) >(6x^2 - 7)/(3x+1)

that as far as i could go
 
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  • #2
Keep making approximations

(6x^2 - 7sinx)/(3x+1) >(6x^2 - 7)/(3x+1) > (6x^2 - 7sinx)/(3x+1) >(6x^2 - 7)/(3x+3x)

(6x^2 - 7)/(3x+3x) = (6x^2 -7)/6x = x -7/(6x)
 
  • #3
(6x^2 - 7sinx)/(3x+1) >(6x^2 - 7)/(3x+3x)=x-7/x
what now
?
 
  • #4
nhrock3 said:
(6x^2 - 7sinx)/(3x+1) >(6x^2 - 7)/(3x+3x)=x-7/x
what now
?

What do you mean "what now "? Are not capable of thinking for yourself ?

If you have something like

x - 7/x if x>M and M > 1008

then

x - 7/x > x -7 > 1008 -7 = 1001
 

FAQ: Proving Inequality: (6x^2 - 7sinx)/(3x+1) >1000

How do you prove this inequality?

In order to prove this inequality, we can use mathematical techniques such as algebraic manipulation and calculus. We will need to show that the left side of the inequality is always greater than the right side for all values of x.

Can you give an example of x that satisfies this inequality?

Yes, for example, if we let x = 1, then the left side of the inequality becomes (6 - 7sin1)/(3+1) = (6 - 0.841)/(3+1) ≈ 5.159/4 ≈ 1.289 which is indeed greater than 1000.

3. Are there any restrictions on the values of x for which this inequality holds true?

Yes, there are restrictions on the values of x. For this inequality to hold, x must be greater than -1/3 and cannot be equal to -1/3. This is because the denominator, 3x+1, cannot equal 0.

4. Can we use a graph to prove this inequality?

Yes, we can use a graph to visually show that the left side of the inequality is always greater than the right side. However, this would not be a formal proof and we would still need to use mathematical techniques to prove it for all values of x.

5. How does this inequality relate to real-world situations?

This inequality may be used in various scientific and engineering fields to model and analyze data. For example, it can be used in economics to compare the cost of two different investments. It can also be used in physics to study the motion of objects in certain systems.

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