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Math Help - some ideas in inequalities that i don't get.

  1. #1
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    some ideas in inequalities that i don't get.

    Hello

    http://im35.gulfup.com/GDxl1.png

    since my mother language is not English, I find it difficult to understand some explanations. and i hope here i could find some help.

    the above picture has a very strange way of solving inequalities !!
    i need full explanation please.
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  2. #2
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    Re: some ideas in inequalities that i don't get.

    Quote Originally Posted by SHEvA9 View Post
    Hello

    http://im35.gulfup.com/GDxl1.png

    since my mother language is not English, I find it difficult to understand some explanations. and i hope here i could find some help.

    the above picture has a very strange way of solving inequalities !!
    i need full explanation please.
    1. You are right: The explanation refers to the inequality

    (x-2)(x+3)>0

    2. Here are some (possible) steps to solve the given inequality:
    • Re-write so you have only one quotient.
    • Determine numerator and denominator so that the complete quotient is positive.


    3. In detail:

    \frac{2x-3}{3x-5}\ge 3~\implies~\frac{2x-3}{3x-5}-3 \ge 0~\implies~\frac{-7x+12}{3x-5}\ge 0

    4. A quotient equals zero or is greater than zero if numerator and denominator have the same sign:

    Case 1: Both are positive: (Be aware that the denominator can't be zero)

    -7x+12\ge 0~\wedge~3x-5 > 0~\implies~x \le \frac{12}7~\wedge~x > \frac53

    That means: \frac53 < x \le \frac{12}7

    Case 2: I'll leave this one for you.
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  3. #3
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    Re: some ideas in inequalities that i don't get.

    thank you a lot
    I agree with what you said
    and i got case 2 as it is the same thing as case 1
    the two factors are negative, the inequality will be reversed when you multiply by negative denominator then reverse it back after dividing by negative sign of the nominator, right?


    one last thing, " 3,000 Solved Problems in Calculus " is a book i am trying to finish so that i could be ready to study Math in Canadian university, is it a good book? i mean not just in terms of the language i also want to improve my math skills, is the book useful for this purpose?
    Last edited by SHEvA9; February 3rd 2013 at 06:55 AM.
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  4. #4
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    Re: some ideas in inequalities that i don't get.

    Quote Originally Posted by SHEvA9 View Post
    thank you a lot
    I agree with what you said
    and i got case 2 as it is the same thing as case 1
    the two factors are negative, the inequality will be reversed when you multiply by negative denominator then reverse it back after dividing by negative sign of the nominator, right?


    one last thing, " 3,000 Solved Problems in Calculus " is a book i am trying to finish so that i could be ready to study Math in Canadian university, is it a good book? i mean not just in terms of the language i also want to improve my math skills, is the book useful for this purpose? Sorry, but I don't know this book. Remember I live in Germany. Look into the Member's list who's living in Canada and ask her/him by PM.
    Honestly I don't understand your answer refering to the 2nd case. This is what I did:

     \frac{2x-3}{3x-5}\ge 3~\implies~\frac{2x-3}{3x-5}-3 \ge 0~\implies~\frac{-7x+12}{3x-5}\ge 0

    That's of course the same as in case 1.

    Now both, the numerator and the denominator, are negative:

    -7x+12\le 0~\wedge~3x-5 < 0~\implies~x \ge \frac{12}7~\wedge~x < \frac53

    That means: x\in \emptyset

    If you draw the graphs of
    f(x)=\frac{2x-3}{3x-5} and y = 3 and x = \frac53 you'll find the set of solutions in the small thick red bar.
    Attached Thumbnails Attached Thumbnails some ideas in inequalities that i don't get.-lsgmenge.png  
    Last edited by earboth; February 4th 2013 at 01:24 AM.
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