When the of two are equal, either the expressions are equal or the opposite of the expressions are equal. Let's look at an example equation.
|ax+b|=|cx+d|
For this equation, there are two possible cases to consider.
ax+b=cx+d or ax+b=- (cx+d)
To solve the given we will write two equations — similar to those above — when we remove the absolute value.
|20+2x|=|4x+4|
lc 20+2x ≥ 0:20+2x = (4x+4) & (I) 20+2x < 0:20+2x = - (4x+4) & (II)
lc20+2x=4x+4 & (I) 20+2x=- (4x+4) & (II)
l20+2x=4x+4 20+2x=-4x-4
l2x=4x-16 2x=-4x-24
l-2x=-16 2x=-4x-24
lx=8 2x=-4x-24
lx=8 6x=-24
lx= 8 x= -4
After solving an absolute value equation, it is necessary to check for . To do this, we substitute the found solutions into the given equation and determine if a true statement is made.
|20+2x|=|4x+4|
|20+2( 8)|? =|4( 8)+4|
|20+16|? =|32+4|
|36|? =|36|
36=36 ✓
Substituting 8 for x in the equation results in a true statement, so x=8 is not an extraneous solution.
|20+2x|=|4x+4|
|20+2({\color{#FF0000}{\text{-}4}})|\stackrel{?}=|4({\color{#FF0000}{\text{-}4}})+4|
|20-8|\stackrel{?}=|\text{-}16+4|
|12|\stackrel{?}=|\text{-}12|
12\stackrel{?}=|\text{-}12|
12=12 ✓
Substituting -4 for x in the equation results in another true statement, so x=-4 is also not extraneous.
