Inequality

[George Saliaris]

I have considered using C-S, triangular inequality but nothing.. I am asked to proove the inequality using 'point distance from a line.'Any ideas?

 

[Deleted member 4993]

I have considered using C-S, triangular inequality but nothing.. I am asked to proove the inequality using 'point distance from a line.'Any ideas?

Can you please "type" your problem in this case. I have difficulty in deciphering your handwriting.

 

[George Saliaris]

abs(3a+4b+2)=5, prove that a^2+b^2+4b+7>=4a.

 

[Dr.Peterson]

You might start by completing the square in the inequality; it represents the exterior of a circle.

Then consider any point on either of the two parallel lines represented by the equation, and find its distance from the center of the circle. It should always be at least equal to the radius.

On the other hand, have you tried doing exactly what it says? Show that the distance from the center of the circle to the line is at least the circle's radius. If you can't do that, at least show us how you have learned to find such a distance.

 

[Deleted member 4993]

abs(3a+4b+2)=5, prove that a^2+b^2+4b+7>=4a.

Hint:

a^2+b^2+4b+7>=4a

a^2+b^2+4b+7- 4a >= 0 ............ complete square

(a - 2)² + (b + 2)² >= 1

 

[George Saliaris]

You might start by completing the square in the inequality; it represents the exterior of a circle.

Then consider any point on either of the two parallel lines represented by the equation, and find its distance from the center of the circle. It should always be at least equal to the radius.

On the other hand, have you tried doing exactly what it says? Show that the distance from the center of the circle to the line is at least the circle's radius. If you can't do that, at least show us how you have learned to find such a distance.

I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.

Hint:

a^2+b^2+4b+7>=4a

a^2+b^2+4b+7- 4a >= 0 ............ complete square

(a - 2)² + (b + 2)² >= 1

Yeh ik that. P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.

P.S : This probem is from a Romanian Mathematics competition.
*Maybe the inequality has to do with 'other chapters' *

 

[Deleted member 4993]

I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.
Yeh ik that. P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.

P.S : This probem is from a Romanian Mathematics competition.
*Maybe the inequality has to do with 'other chapters' *

What does the following mean geometrically:

abs(3a+4b+2)=5

 

[Dr.Peterson]

I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.

P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.

Do you not see the relationship between knowing the center of the circle and the rest of the problem? Any point farther from the center of the circle than its radius will satisfy the inequality. You can use the formula for the distance from a point to a line to show that every point on the line is farther than the radius. So if you do what I suggested, you will be doing what you were told to do, as I understand it. I think you will like the result. But in order to do that, you have to start! What work have you done?

I have to say that there are typically many ways to prove something, so it is awkward to say that only one method of solution will be accepted. Can you tell us exactly what your instructions were? Also, please show us, as I asked, the formula you refer to, as there may be more than one form for it.

 

[George Saliaris]

The rest is not readable... *Edit* I assumed that 3a+4b+2=0,contradiction...

 

[Dr.Peterson]

Please explain in words what you are doing. You've used symbols without defining them (namely M and e or epsilon), and haven't said what the equations you wrote mean.

I think you may have a correct answer; I just can't be sure! You certainly need to say more to call it a proof.

 

[George Saliaris]

I have not done any significant steps so far...So, I will let you know when I will have.

 

[George Saliaris]

I think I made things even harder...

 

[Dr.Peterson]

You've certainly made it harder to follow what you are doing. Can you at least use a word occasionally to say what it means? And just show us one line of reasoning at a time.

But I asked you merely to explain what you did in post #9, which should not be hard. That may be the core of a proof, once you explain it!

But here's one other question: When you say you are to use the formula for the distance from a point to a line, are you allowed to interpret that as meaning that this is the shortest distance from the point to any point on the line? That turns that formula into an inequality you should be able to use.