Results 1 to 3 of 3

Math Help - Radical inequality

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    65

    Radical inequality

    Greetings,
    I have the below problem, which goes relatively fine up to where I find the roots. I guess every two roots are to form a multitude. I get the two logarithm ones fine, x varies from the smaller log. to the larger one.
    I am unsure what happens when I have no x4. The answer is that x ranges from minus infinity to the other root. However, I do not understand why.
    Attached Thumbnails Attached Thumbnails Radical inequality-ineq.gif  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,740
    Thanks
    646
    Hello, Logic!

    I don't see an inequality . . . Is it "greater than"?
    Also, why are you looking for four roots?


    8\sqrt{2^{2x-2} - 2^x + 1} \;\;{\color{red}>}\;\;2^{2x} - 2^{x+2} + 7
    Those initial steps were excellent! . . . 4\sqrt{2^{2x} - 4\!\cdot\!2^x + 4} \;>\;\left(2^{2x} - 4\!\cdot\!2^x + 4\right) + 3


    Here's my variation of the solution . . .

    Note that: . 2^{2x} - 4\!\cdot\!2^x + 4 \:=\:\left(2^x - 2\right)^2

    So we have: . 4\sqrt{(2^x-2)^2} \;>\;(2^x-2)^2 + 3 \quad\Rightarrow\quad 4(2^x - 2) \;>\;(2^x-2)^2 + 3


    Let t \:=\:2^x-2

    We have: . 4t \;>\;t^2 +3 \quad\Rightarrow\quad -t^2 + 4t - 3 \;>\;0

    Multiply by -1: . t^2 - 4t + 3 \;<\;0 \quad\Rightarrow\quad (t-1)(t-3) \;<\;0

    And we have: . \begin{array}{cccccc} t-1 \:>\:0 & \Rightarrow & t \:>\:1 \\ t-3 \:<\:0 &\Rightarrow& t \:<\:3\end{array}\quad\Rightarrow\quad t \in (1,3)

    Back-substitute: . \begin{array}{ccccccc}2^x-2\:=\:1 & \Rightarrow& 2^x \:=\:3 &\Rightarrow& x \:=\:\log_23 \\ 2^x-2 \:=\:3 &\Rightarrow& 2^x \:=\:5 &\Rightarrow& x \:=\:\log_25 \end{array}


    Therefore: . \log_23 \:<\:x\:<\:\log_25

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2008
    Posts
    65
    Yes, yes, it is greater than.

    Your solution seems very nice, too, however this is not the full answer.
    It is said that every number from the interval minus infinity to zero also works, and, indeed, for -1, for example, I get 24 > 21 and -1 is not from <br /> <br />
\log_23 \:<\:x\:<\:\log_25<br />
right.. ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Ring isomorphism maps a Jacobson radical into a Jacobson radical?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: June 21st 2010, 10:45 AM
  2. Radical-Radical equation
    Posted in the Algebra Forum
    Replies: 6
    Last Post: June 27th 2009, 03:38 AM
  3. Inequality cubic-radical
    Posted in the Algebra Forum
    Replies: 1
    Last Post: June 19th 2009, 04:13 PM
  4. Radical inequality
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: June 17th 2009, 03:25 AM
  5. Radical inside a radical?
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 14th 2007, 09:59 PM

Search Tags


/mathhelpforum @mathhelpforum