How do you solve and check for extraneous solutions in #abs(2t-3) = t#?

Question

1307 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-08-01T15:22:22

#color(red)(t = 3)# is a solution.
#color(red)(t = 1)# is an extraneous solution.

SOLVE

#|2t-3| = t#

We need to write two different equations without the absolute value symbols and solve for #t#.

These equations are:

(1): #(2t-3) = t#
(2): #-(2t-3) = t#

Solve Equation 1:

#2t-3 = t#

Subtract #t# from each side.

#t-3 = 0#

Add #3# to each side.

#t = 3#

Solve Equation 2:

#−(2t-3) = t#

Remove parentheses.

#-2t+3= t#

Add #2t# to each side.

#3 = 3t#

Divide each side by #3#.

#t = 1#

The solutions are #t = 1# and #t = 3#.

CHECK FOR EXTRANEOUS SOLUTIONS:

If #t = 1#,

#|2t-3|=t#
#|2(1)-3|= 3#
#|2-3| = 5#
#|-1| =5#
#1=5#

This is impossible, so #t=1# is an extraneous solution.

If #t = 3#,

#|2t-3| = t#
#|2(3) - 3| = 3#
#|6 - 3| = 3#
#|3| = 3#
#3 = 3#

#t=3# is a solution.