Solve:
#(t+5)/2+t/6=(t-6)/8-8#
In order add or subtract fractions, they must have the same denominator, called the least common denominator (LCD). List the multiples of #2,6, and 8#.
#2:##2,4,6,8,10,12,14,16,18,20,22,color(red)24,26,28.....#
#6:##6,12,18,color(red)24,30,36.....#
#8:##8,16,color(red)24,32.....#
The LCD is #color(red)24#.
Now each fraction will need to be multiplied by an equivalent fraction that will result in a denominator of #color(red)24#. An #color(green)("equivalent fraction")# is equal to #color(green)1#. For example, #color(green)(2/2=1)#.
#(t+5)/2xxcolor(green)(12/12)+t/6xxcolor(green)(4/4)=(t-6)/8xxcolor(green)(3/3)-8/1xxcolor(green)(24/24)#
Simplify.
#(12(t+5))/24+(4t)/24=(3(t-6))/24-192/24#
#color(blue)(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~#
Alternatively, you can skip multiplying by equivalent fractions, and multiply all terms by the LCD #24# directly.
#color(blue)(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~#
Multiply each term by #24#.
#24((12(t+5))/24+(4t)/24=(3(t-6))/24-192/24)#
Simplify.
#12(t+5)+4t=3(t-6)-192#
Expand.
#12t+60+4t=3t-18-192#
Simplify.
#16t+60=3t-210#
Subtract #3t# from both sides.
#16t-3t+60=color(red)cancel(color(black)(3t))-210-color(red)cancel(color(black)(3t))#
Simplify.
#13t+60=-210#
Subtract #60# from both sides.
#13t+color(red)cancel(color(black)(60))-color(red)cancel(color(black)(60))=-210-60#
Simplify.
#13t=-270#
Divide both sides by #13#.
#t=-270/13=-20 10/13#
