1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inequalities problem

  1. Apr 25, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the set of all ##x## for which ##\dfrac{2x}{2x^2 + 5x + 2} > \dfrac{1}{x + 1}##

    2. Relevant equations


    3. The attempt at a solution


    I'm getting two different sets of answers with two different methods:


    Method 1-Wrong

    ##\dfrac{2x}{2x^2 + 5x + 2} > \dfrac{1}{x + 1}##


    ##\dfrac{2x^2 + 5x + 2}{2x} < x + 1##


    ##\dfrac{2x^2 + 5x + 2}{2x} - (x+1) < 0##


    ##\dfrac{2x^2 + 5x + 2 - 2x(x+1)}{2x} < 0##


    ##\dfrac{3x + 2}{2x} < 0##


    ## x \in \left( \dfrac{-2}{3}, 0 \right)##



    Method 2, the correct one

    ##\dfrac{2x}{2x^2 + 5x + 2} > \dfrac{1}{x + 1}##


    ##\dfrac{2x}{2x^2 + 5x + 2} - \dfrac{1}{x + 1} > 0##


    ##\dfrac{2x(x+1) - (2x^2 + 5x + 2)}{(2x^2 + 5x + 2)(x + 1)} > 0##


    ##\dfrac{3x + 2}{(2x + 1)(x + 1)(x + 2)} < 0 ##


    ##x \in (-2,-1) \cup \left(\dfrac{-2}{3} , \dfrac{-1}{2}\right)##
     
  2. jcsd
  3. Apr 25, 2016 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    In method 1, the first step is correct only if the left hand side and right hand side have the same sign.

    If the left hand side is positive and the right hand side is negative, then which way does the inequality go after taking the reciprocal?
     
  4. Apr 25, 2016 #3
    Oh! Got it, thank you :)
     
  5. Apr 25, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    How do you know you can go from ##\dfrac{2x}{2x^2 + 5x + 2} - \dfrac{1}{x + 1} > 0## to ##\dfrac{2x(x+1) - (2x^2 + 5x + 2)}{(2x^2 + 5x + 2)(x + 1)} > 0##? Certainly, if you multiply both sides of an inequality by a positive quantity, the inequality remains unchanged in direction. However, if you multiply both sides by a negative, the inequality is reversed.
     
  6. Apr 25, 2016 #5

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    It looks more like OP combined rational expressions by using a common denominator, rather than by multiplying the whole expression by anything.
     
  7. Apr 25, 2016 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Agreed, but I would have preferred that the OP answer the question.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Inequalities problem
  1. Inequality problem (Replies: 14)

  2. Inequality problem (Replies: 11)

  3. Inequality problem (Replies: 2)

  4. Inequality problem (Replies: 1)

  5. Inequality problem (Replies: 11)

Loading...