Math problem - help please

Discussion in 'Secondary Education' started by Upsadaisy, May 15, 2012.

  1. Upsadaisy

    Upsadaisy Moderator

    Joined:
    Aug 2, 2002
    Messages:
    18,938
    Likes Received:
    681

    May 15, 2012

    Would you help me solve this?

    x/(x + 1) + 3/(x + 4) = (x + 3)/(x + 4)

    Thanks so much.
     
  2.  
  3. Mathemagician

    Mathemagician Groupie

    Joined:
    Feb 4, 2011
    Messages:
    1,372
    Likes Received:
    0

    May 15, 2012

    Let's get a common denominator for the LHS! Just like if you had 2/3+1/5, you would have an LCD of 3*5=15, let's take the same approach. Here our LCD is (x+1)(x+4).

    So to make the denominator of each (x+1)(x+4), I think of it as multiplying both terms on the left by (x+1)(x+4)/(x+1)(x+4).

    So x/(x+1) becomes x(x+4)/(x+1)(x+4) and 3/(x+4) becomes 3(x+1)/(x+1)(x+4).

    Now we can combine the like terms on the LHS since the denominator is the same.

    So we have

    (x*(x+4)+3*(x+1))/(x+1)(x+4)=(x+3)/(x+4).

    Now we can multiply both sides by x+4 since we see we can get some nice cancellation.

    We have (x*(x+4)+3*(x+1)/(x+1)=x+3.

    Now multiply both sides by (x+1), expand, and have a field day:

    (x^2+4x+3x+3)=x^2+4x+3

    This simplifies to 3x=0.

    So x=0.

    We plug in, and see that it is indeed correct.

    Wolfram Alpha verifies it:

    http://www.wolframalpha.com/input/?i=(x/(x+1))+3/(x+4)=(x+3)/(x+4)

    Does that make sense? Let me know if something is unclear.
     
  4. mopar

    mopar Multitudinous

    Joined:
    Aug 15, 2010
    Messages:
    10,924
    Likes Received:
    0

    May 15, 2012

    I was thinking a little differently:

    You could subtract the 3/(x+4) to the other side of the equation, so you have:
    x/(x+1) = (x+3)/(x+4) - 3/(x+4)

    Then combine the two fractions on the right:
    x/(x+1) = (x+3-3)/(x+4)

    Then I would simplify:
    x/(x+1) =x/(x+4)

    Cross multiply:

    x(x+4) = x(x+1)

    Simplify and solve:
    x^2 + 4x = x^2 + x

    3x=0, so x=0.
     
  5. Upsadaisy

    Upsadaisy Moderator

    Joined:
    Aug 2, 2002
    Messages:
    18,938
    Likes Received:
    681

    May 15, 2012

    That is perfectly clear, Mathemagician. Thank you so much. I guess I just needed a magician to clear things up. I had somehow messed up the denominators along the way... I appreciate your help.
     
  6. Upsadaisy

    Upsadaisy Moderator

    Joined:
    Aug 2, 2002
    Messages:
    18,938
    Likes Received:
    681

    May 15, 2012

    That is really interesting, mopar. Thank you for that insight. Did you jump to that solution right away because of the x + 4 denominator on both sides? I was so fixated on getting a LCD that I didn't even think of that.
     
  7. mopar

    mopar Multitudinous

    Joined:
    Aug 15, 2010
    Messages:
    10,924
    Likes Received:
    0

    May 15, 2012

    As soon as I saw the x+4 on both sides, I thought it would be much easier than trying to deal with multiply by (x+1)(x+4) to three fractions.
     
  8. Mathemagician

    Mathemagician Groupie

    Joined:
    Feb 4, 2011
    Messages:
    1,372
    Likes Received:
    0

    May 15, 2012

    Np, mopar has a much more efficient solution actually! Sometimes with math you can't see the forest through the trees. When I see something like this, I am so quick to find a common denominator that easy tricks slip by me. At least we know both methods work haha!
     
  9. Upsadaisy

    Upsadaisy Moderator

    Joined:
    Aug 2, 2002
    Messages:
    18,938
    Likes Received:
    681

    May 15, 2012

    My student will have to know how to do it the way you presented, though, so I'm glad you did so.
     
  10. HMM

    HMM Cohort

    Joined:
    Dec 21, 2004
    Messages:
    694
    Likes Received:
    1

    May 15, 2012

    mopar...I would do this..

    x/(x + 1) = x/(x + 4)

    from here x=0 is clearly a solution

    so now we can cancel the x's

    1/(x + 1) = 1/(x + 4)

    reciprocate

    x+1=x+4
    1=4

    thus there an no other solutions. ie the only solution is x=0
     

Share This Page

Members Online Now

  1. Styleons USA,
  2. Backroads
Total: 411 (members: 3, guests: 379, robots: 29)
test