1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Help with algebraic manipulation of an inequality

  1. Oct 3, 2008 #1
    Let R denote the set of all real numbers and Q the set of all rational numbers.

    Statement to prove:
    If x and y are in R with x < y, show that x < ty + (1-t)x < y
    for all t, 0 < t < 1.

    My work on the proof so far:

    Given x and y are real numbers with x < y. By theorem we know there exists an r in Q such that x < r < y. Take r = t. So x < t < y.

    That's all I have so far.

    My professor said this proof was more of algebraic manipulation. I am stuck as to how I can algebraically manipulate the inequality to get to x < ty + (1-t)x < y.
  2. jcsd
  3. Oct 3, 2008 #2
    ok , here it is what i think

    since y>x=> y-x>0, now since 0<t<1, we have

    0<t(y-x)<y-x add an x on both sides and we get




    what we actually need to show
  4. Oct 3, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    If x<y then xt+x(1-t)<yt+x(1-t)<yt+y(1-t).
  5. Oct 3, 2008 #4
    thanks for all the replies. i will try all these and reply back when i've gotten more work done on my own.

    thank you very much!
  6. Oct 5, 2008 #5
    Awesome! Thank you very much.:smile: That was very clear. I totally understood it and get it now! :cool:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook