MathYouLike MathYouLike
  • Toggle menubar
  • Toggle fullscreen
  • Toggle Search
  •    Sign up
  • QUESTIONS
  • TAGS
  • USERS
  • BADGES
  • UNANSWERD
  • ASK A QUESTION
  • BLOG
...

Kayasari Ryuunosuke Coordinator

21/07/2017 at 21:56
Answers
2
Follow

Find x know :

|x - 1| + |2x - 4| = 3




    List of answers
  • ...
    Phan Thanh Tinh Coordinator 22/07/2017 at 09:36

    1st case : \(x< 1\Leftrightarrow x-1< 0;2x-4< -2\)

    \(\Rightarrow\left|x-1\right|=1-x;\left|2x-4\right|=4-2x\).We have :

    \(1-x+4-2x=3\Leftrightarrow5-3x=3\Leftrightarrow3x=2\Leftrightarrow x=\dfrac{2}{3}\)(satisifed)

    2nd case : \(1\le x< 2\Leftrightarrow x-1\ge0;2x-4< 0\)

    \(\Rightarrow\left|x-1\right|=x-1;\left|2x-4\right|=4-2x\).We have :

    \(x-1+4-2x=3\Leftrightarrow3-x=3\Leftrightarrow x=0\) (absurd)

    3rd case : \(x\ge2\Rightarrow x-1\ge1;2x-4\ge0\)

    \(\Rightarrow\left|x-1\right|=x-1;\left|2x-4\right|=2x-4\).We have :

    \(x-1+2x-4=3\Leftrightarrow3x-5=3\Leftrightarrow3x=8\Leftrightarrow x=\dfrac{8}{3}\)(satisfied)

    So,\(S=\left\{\dfrac{2}{3};\dfrac{8}{3}\right\}\)


     

    Selected by MathYouLike
  • ...
    Aim Egst 21/07/2017 at 23:52

    \(\left|x-1\right|+\left|2x-4\right|=3\)

    \(\Leftrightarrow\sqrt{\left(x-1\right)^2}+\sqrt{\left(2x-4\right)^2}=3\)

    \(\Leftrightarrow\sqrt{x^2-2x+1}+\sqrt{4x^2-16x+16}=3\)

    \(\Leftrightarrow\sqrt{x^2-2x+1}-\left(\dfrac{2}{3}x-\dfrac{1}{9}\right)+\sqrt{4x^2-16x+16}-\left(-\dfrac{2}{3}x+\dfrac{28}{9}\right)=0\)

    \(\Leftrightarrow\dfrac{x^2-2x+1-\left(\dfrac{2}{3}x-\dfrac{1}{9}\right)^2}{\sqrt{x^2-2x+1}+\dfrac{2}{3}x-\dfrac{1}{9}}+\dfrac{4x^2-16x+16-\left(-\dfrac{2}{3}x+\dfrac{28}{9}\right)^2}{\sqrt{4x^2-16x+16}+\left(-\dfrac{2}{3}x+\dfrac{28}{9}\right)}=0\)

    \(\Leftrightarrow\dfrac{\dfrac{45x^2-150x+80}{81}}{\sqrt{x^2-2x+1}+\dfrac{2}{3}x-\dfrac{1}{9}}+\dfrac{\dfrac{288x^2-960x+512}{81}}{\sqrt{4x^2-16x+16}+\left(-\dfrac{2}{3}x+\dfrac{28}{9}\right)}=0\)

    \(\Leftrightarrow\dfrac{\dfrac{5\left(3x-8\right)\left(3x-2\right)}{81}}{\sqrt{x^2-2x+1}+\dfrac{2}{3}x-\dfrac{1}{9}}+\dfrac{\dfrac{32\left(3x-8\right)\left(3x-2\right)}{81}}{\sqrt{4x^2-16x+16}+\left(-\dfrac{2}{3}x+\dfrac{28}{9}\right)}=0\)

    \(\Leftrightarrow\dfrac{\left(3x-8\right)\left(3x-2\right)}{81}\left(\dfrac{5}{\sqrt{x^2-2x+1}+\dfrac{2}{3}x-\dfrac{1}{9}}+\dfrac{32}{\sqrt{4x^2-16x+16}-\dfrac{2}{3}x+\dfrac{28}{9}}\right)=0\)

    \(\Rightarrow\left[{}\begin{matrix}3x-8=0\\3x-2=0\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{8}{3}\\x=\dfrac{2}{3}\end{matrix}\right.\)

    Kayasari Ryuunosuke selected this answer.

Post your answer

Please help Kayasari Ryuunosuke to solve this problem!



Weekly ranking


© HCEM 10.1.29.225
Crafted with by HCEM