How do you solve #|2x - 3| = 7#?

We know that if #absa = b#, #a = +-b#.

First, remove the absolute value. Since we do not know if #2x - 3# is positive or negative, we will need a #+-# on the #7#:

#abs(2x - 3) = 7 => 2x-3 = +- 7#

Then, add #3# to both sides, and finally divide by #2#:

#2x = 3 +- 7#

#x = 3/2 +- 7/2#

So #x# is equal to either #10/2 = 5# or #-4/2 = -2#.

Because this equation contains the absolute value function there will be two solutions to this problem.

The absolute value function takes any negative or positive term and transforms it its positive value.

Therefore we must solve the term with in the absolute value function for both the positive and negative form of its equal.

Solution 1)

#2x - 3 = 7#

#2x - 3 + color(red)(3) = 7 + color(red)(3)#

#2x - 0 = 10#

#2x = 10#

#(2x)/color(red)(2) = 10/color(red)(2)#

#(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = 5#

#x = 5#

Solution 2)

#2x - 3 = -7#

#2x - 3 + color(red)(3) = -7 + color(red)(3)#

#2x - 0 = -4#

#2x = -4#

#(2x)/color(red)(2) = -4/color(red)(2)#

#(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = -2#

#x = -2#

The two solutions to this problem are:

#x = 5# and #x = -2#