Expert: Paul Klarreich - 12/31/2011

Question
Hi Mr. Klarreich,

I need help with this maths question:

|t+2|+|3t-1|<5

Thank you

Answer
Questioner:Rima
Country:New South Wales, Australia
Category:Advanced Math
Private: No   <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< CHANGED
Subject:Maths Inequalities/Absolute Value
Question:

Hi Mr. Klarreich,

I need help with this maths question:

|t+2|+|3t-1|<5


You have to 'interpret' the absolute value symbols:

| t + 2 |  'breaks' at t = -2:

When  t < -2,  you write -(t+2)
When  t >= -2,  you write (t+2)


| 3t - 1 |  'breaks' at t = 1/3
When  t < 1/3,  you write -(3t-1)
When  t >= 1/3,  you write (3t-1)

So you will want to have THREE separate inequalities:
         -2         -1          0    1/3
-----------+-----------X-----------+-----------+---X-------+-----------+
-(t+2) - (3t-1) < 5        (t+2) - (3t-1) < 5          (t+2) + (3t-1) < 5
<<<<<< FIRST>>>>>>>>>><<<<<  SECOND >>>>>>>>>>>>>><<<< THIRD >>>>>>>>>>>>>>>


Solve each and check consistency:

FIRST:  - t - 2 - 3t + 1 < 5
         - 4t - 1 < 5  
         - 4t < 6
         t > -3/2   <<< Flip to > when dividing by (-4)
Now t > -3/2 is not consistent with t < -2, so this gives no solutions.

SECOND:  (t+2) - (3t-1) < 5
        t + 2 - 3t + 1 < 5
         - 2t + 3 < 5
         - 2t < 2
         t > - 1   >> flip.
t > -1 is consistent with  -2 < t < 1/3, so you have  -1 < t < 1/3 as some solutions.


Third:   t + 2 + 3t - 1 < 5
         4t + 1 < 5
         4t < 4
         t < 1

Consistent:

Your solutions look like:

         -2         -1        x+2)   0    1/3
-----------+-----------X-----------+-----------+---X-------+-----------+
         *************************

If you use your graphing calculator, you'll see it.