How do you solve #abs(7 + 2x) = 9#?

As The Absolute Value of the Expression is 9, we will have to solve the equation twice, once for positive and once for negative.

As. #|a| = a = |-a|#.

So, Case 1 (Taking Positive):

#color(white)(xxx)(7 + 2x) = 9#

#rArr cancel7 + 2x cancel(- 7) = 9 - 7# [Subtract #7# from both sides]

#rArr 2x = 2#

#rArr (2x)/2 = 2/2# [Dividing both sides by #2#]

#rArr x = 1#

Case 2 (Taking Negative) :

#color(white)(xxx)-(7 + 2x) = 9#

#rArr - 7 - 2x = 9# [Distributive Property]

#rArr cancel(-7) - 2x + cancel7 = 9 + 7# [Add #7# to both sides]

#rArr -2x = 16#

#rArr (-2x)/-2 = 16/-2# [Dividing both sides by #-2#]

#rArr x = -8#

So, #x# has two values, #1, -8#.

Hope this helps.

#"the expression inside the absolute value bars can be"#
#"positive or negative so there are 2 possible solutions"#

#color(magenta)"Positive expression"#

#7+2x=9#

#"subtract 7 from both sides and divide by 2"#

#rArr2x=9-7=2rArrx=2/2=1#

#color(magenta)"Negative expression"#

#-(7+2x)=9#

#rArr-7-2x=9#

#"add 7 to both sides and divide by "-2#

#rArr-2x=9+7=16rArrx=16/(-2)=-8#

#color(blue)"As a check"#

Substitute these values into the left side of the equation and if equal to the right side then they are the solutions.

#x=1to|7+2|=|9|=9#

#x=-8to|7-16|=|-9|=9#

#rArrx=-8" or "x=1" are the solutions"#