How do you solve #abs(x+3)=abs(7-x)#?

1 Answer
Mar 30, 2015

One way(Method 1) would be to replace each of the absolute value expressions with 2 possible expressions (one positive and one negative). For the given example there are two absolute value expression and all 4 possible combinations would need to be considered.

An alternative (Method 2) would be to square both sides and solve.

Method 1
#-(x+3) = -(7-x)#
#rarr -2x = -4#
#rarr x = 2#

#-(x+3) = +(7-x)#
#rarr -3 = 7# impossible; extraneous result; ignore

#+(x+3) = -(7-x)#
#rarr 3=-7# Impossible; extraneous result; ignore

#+(x+3) = +(7-x)#
#rarr 2x = 4#
#rarr x = 2# (a duplicate of the second result)

The only solution is #x=2#

Method 2
#(abs(x+3))^2 = (abs(7-x))^2#

#cancel(x^2) +6x + 9 = 49 -14x +cancel(x^2)#
#rarr 20x = 40#
#x=2#