Let's start from definition of an absolute value: The absolute value of "a" is "a" itself, if "a" is equal to or grater than 0 (other words: positive or zero), and "-a" if "a" is negative. For example l5l = 5, l-5l = -(-5)=5.
In the equation ld-3l=2d+9 there is variable inside the absolute value, therefore the expression (d-3) can be positive, can be negative.
1. Let's assume that (d-3) is positive and we will ignore the sign of absolute value and will rewrite original equation as "d-3=2d+9" . What does it mean "to solve equation"? We have to leave variable by itself, if there are variables and numbers on both sides of equation, move all variables onto one side of equation and numbers onto another side. In order to do so, we will subtract 2d from both sides of equation and add 3 to the both sides.
Now, there will be "d-2d=9+3", simplify (add/subtract like terms):
"-d=12" multiply both sides by (-1), remember that variable should be by itself
(no number, no signs) "d=-12"
2. Let's assume that (d-3) is negative, then we have to rewrite the equition with no absolute value signs but with "-" before (d-3), to make this expression positive.
There will be "-(d-3)=2d+9", open parentheses, remember if there is minus before parentheses we have to change the sign of each term inside to opposite "-d+3=2d+9" . Subtract "2d" and "3" from both sides, combine like terms, and we have "-3d=6", So "d=-2"
Last and very important step is we have to check our answer: first let's replace "d" by "-12" in original equation:
15=-15 the statement is false therefore "-12" is not the root of original equation
Let's replace "d" by "-2"
5=5 statement is true, therefore "-2" is the only root of the original equation