Notice that, #a=|b|=>a=+-b#.
In other words, there are two possibilities, either the expression inside the absolute bars is positive, or it is negative.
If it is positive, then we can just remove the absolute bars: #5x+24=8-3x#. Solving this gives #8x=-16# and #x=-2#. We make sure that this solution gives the expression inside the absolute bars a positive value: #8-3x=8-3*-2=14>0#.
If it is negative, then we can remove the absolute bars and negate it: #-(5x+24)=8-3x#. Solving this gives #2x=-16# and #x=-8#. We make sure that this solution gives the expression inside the absolute bars a negative value: #8-3x=8-3*-8=32>0#. This means that this solution is incorrect because we assumed that the expression inside the absolute bars is negative.
The reason that this is wrong can be seen by looking at the equation we were trying to solve: #-(5x+24)=8-3x#. Since #8-3x# is positive, #5x+24# must be negative. However, as seen in the original equation, it is equal to the absolute value of some number, which is always positive.
So, there is only one possible solution, #x=-2#. It is always wise to check that the solutions are correct for equations like these.
