- #1

bamajon1974

- 21

- 5

- Homework Statement
- For context, this question is not for homework but in reference to bounding terms for epsilon-delta limit proofs. I know you can bound the numerator and denominator separately but I really would like to know how manipulate the quotient together as I have forgotten some elementary algebra.

- Relevant Equations
- Please see below.

How does one manipulate rational absolute inequalities?

For example, I want to transform the absolute value inequality ##|x-3|<1## to ##\frac{|x+3|}{5x^2}<A \ ##, for some number ##\text{A}##, to find an upper and lower bound on the latter term using the constraint in the first term, and not sure what to do with the denominator and changing inequality direction.

I can expand the absolute value inequality as follows: ##|x-3|<1 \implies -1<x-3<1 \implies 2<x<4 \implies 5<x+3<7. ## How do I introduce the ##5x^2## term in the denominator? Dividing all three sides by ##5x^2## would add a variable to the numbers and possibly change signs. Not sure where to go from here.

For context, this question is not for homework but in reference to bounding terms for epsilon-delta limit proofs. I know you can bound the numerator and denominator separately but I really would like to know how manipulate the quotient together as I have forgotten some elementary algebra.

Thanks!

For example, I want to transform the absolute value inequality ##|x-3|<1## to ##\frac{|x+3|}{5x^2}<A \ ##, for some number ##\text{A}##, to find an upper and lower bound on the latter term using the constraint in the first term, and not sure what to do with the denominator and changing inequality direction.

I can expand the absolute value inequality as follows: ##|x-3|<1 \implies -1<x-3<1 \implies 2<x<4 \implies 5<x+3<7. ## How do I introduce the ##5x^2## term in the denominator? Dividing all three sides by ##5x^2## would add a variable to the numbers and possibly change signs. Not sure where to go from here.

For context, this question is not for homework but in reference to bounding terms for epsilon-delta limit proofs. I know you can bound the numerator and denominator separately but I really would like to know how manipulate the quotient together as I have forgotten some elementary algebra.

Thanks!

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