Results 1 to 3 of 3

Thread: inequality

  1. #1
    Senior Member
    Jan 2009


    Find the solutions set of the inequality $\displaystyle |x-2|<\frac{1}{x}$ , where x is not 0 .

    THanks .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Mar 2007
    Santiago, Chile
    it's quite obvious that $\displaystyle x\ne0,$ that's an assumption that it must be implicit and that you should know.

    well then, by a well known property your inequality equals $\displaystyle -\frac{1}{x}<x-2<\frac{1}{x},$ so you have to solve each one.

    i'll help ya with the first one:

    $\displaystyle \begin{aligned}
    x-2&>-\frac{1}{x} \\
    x-2+\frac{1}{x}&>0 \\

    the numerator is a perfect square, but it has a critical point at $\displaystyle x=1,$ but we don't actually want that, since that'd turn the numerator zero and the inequality won't be satisfied, so to solve the previous inequality we just require that $\displaystyle x>0,$ but since we've stated that $\displaystyle x\ne1,$ the first solution set is $\displaystyle (0,1)\cup(1,\infty).$

    apply the same procedure for the second inequality, that'd yield a second solution set, which you need to intersect with the first one to get the full solution set.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor red_dog's Avatar
    Jun 2007
    Medgidia, Romania
    If $\displaystyle x<0$, the inequality has no solution.

    If $\displaystyle x>0$ the inequality is $\displaystyle x|x-2|<1$

    1) $\displaystyle x\in(0,2)\Rightarrow x(2-x)<1\Rightarrow (x-1)^2>0$, which is true.

    2) $\displaystyle x\in[2,\infty)\Rightarrow x^2-2x-1<0\Rightarrow x\in(1-\sqrt{2},1+\sqrt{2})\cap [2,\infty)\Rightarrow x\in[2,1+\sqrt{2})$

    From 1) and 2), $\displaystyle x\in(0,2)\cup[2,1+\sqrt{2})\Rightarrow x\in(0,1+\sqrt{2})$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Jan 11th 2011, 08:20 PM
  2. Replies: 3
    Last Post: Dec 12th 2010, 01:16 PM
  3. inequality
    Posted in the Math Challenge Problems Forum
    Replies: 7
    Last Post: Jul 25th 2010, 06:11 PM
  4. Inequality help
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: Jul 8th 2010, 06:24 AM
  5. Inequality :\
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Oct 12th 2009, 01:57 PM

Search Tags

/mathhelpforum @mathhelpforum