Solve the equation. check for extraneous solutions.

Assume
if |a|=b
a=b and -a=b

so

9|9-8x|=2x+3
divide both sides by 9
|9-8x|=2/9x+1/3
assume
9-8x=2/9x+1/3 and
-9+8x=2/9x+1/3

9-8x=2/9x+1/3
times 9 both sides
81-72x=2x+3
add 72x both sides
81=74x+3
minus 3
78=74x
divide both sides by 74
39/37=x

other one
-9+8x=2/9x+1/3
multiply both sides by 9
-81+72x=2x+3
minus 2x both sides
-81+70x=3
add 81 to both sides
70x=84
divide boh sides by 70
x=6/5

x=39/37 and 6/5
check

9|9-8(39/37)|=2(39/37)+3
2646/37=189/37
false

9|9-8(6/5)|=2(6/5)+3
351/5=27/5
false

no soltion evidently, weird