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!

Absolute Value Equations: Setting Restrictions

  1. Jan 9, 2008 #1
    Hello everyone,

    For the following absolute value equations, I have no trouble solving them and finding the valid [tex] x [/tex] solutions by plugging all the x solutions into the original equation.

    However, I am just wondering if could someone please show me how to set restrictions for the following equations? I want to know how to solve these equations in two ways instead of just one.

    Thank you very much.

    ---

    1. [tex] | 2|x + 3| - 5 | = 7 [/tex]

    2. [tex] | x - | 2x + 1 || = 3 [/tex]
     
  2. jcsd
  3. Jan 10, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The first one isn't an "absolute value" equation at all since the variable, x, does not appear in the absolute vaues: |2|= 2 and |-5|= -5 so that equation is just 2x+ 15= 7.

    For the second one, |2x+ 1| "changes" when 2x+ 1= 0 or x= -1/2. Now separate it into cases:

    If x< -1/2, then 2x+ 1< 0 so |2x+1|= -(2x+1)= -2x-1. Your equation is now |x+ 2x+ 1|= |3x+1|= 3. |3x+1| "changes" when 3x+ 1= 0 or x= -1/3. Since -1/3< -1/2, x< -1/2 immediately gives x< -1/3 so we have -(3x-1)= -3x-1= 3. Solve that equation. If the solution is less than -1/2, that is a solution to the original problem. If not, this gives no solution.

    If x>= -1/2, then 2x+ 1>= 0 so |2x+1|= 2x+ 1. Your equation is now |x- 2x- 1|= |-3x-1|= 3. |-3x-1| "changes" when -3x- 1= 0 or x= -1/3. Now we have two possibilities: -1/2<= x< -1/3 or -1/3<= x.
    If -1/2<= x< -1/3, then |-3x-1|= -(-3x-1)= 3x+ 1 and the equation becomes 3x+ 1= 3. Solve that for x and check if the answer is between -1/2 and -1/3. If it is, that is a solution to the original equation, if not, this gives no solution.

    Finally, if x>= -1/3 then |-3x-1|= -3x-1 and the equation become -3x-1= 3. Solve that for x and check if the answer is >= -1/3.
     
  4. Jan 11, 2008 #3
    halls i think you misinterpreted the first equation, it's a little ambiguous. he probably meant |2(|x+3|)-5|.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Absolute Value Equations: Setting Restrictions
  1. Absolute value (Replies: 1)

Loading...