First, multiply each side of the equation by #color(red)(12)# to eliminate the fractions while keeping the equation balanced:

#color(red)(12)((x - 1)/4 - x/3) = color(red)(12) xx 1/12#

#(color(red)(12) xx (x - 1)/4) - (color(red)(12) xx x/3) = cancel(color(red)(12)) xx 1/color(red)(cancel(color(black)(12)))#

#(cancel(color(red)(12))3 xx (x - 1)/color(red)(cancel(color(black)(4)))) - (cancel(color(red)(12))4 xx x/color(red)(cancel(color(black)(3)))) = 1#

#3(x - 1) - 4x = 1#

Next, expand the terms in parenthesis:

#(3 xx x) - (3 xx 1) - 4x = 1#

#3x - 3 - 4x = 1#

Then, group and combine like terms on the left side of the equation:

#3x - 4x - 3 = 1#

#(3 - 4)x - 3 = 1#

#-1x - 3 = 1#

Next, add #color(red)(3)# to each side of the equation to isolate the #x# term while keeping the equation balanced:

#-1x - 3 + color(red)(3) = 1 + color(red)(3)#

#-1x - 0 = 4#

#-1x = 4#

Now, multiply each side of the equation by #color(red)(-1)# to solve for #x# while keeping the equation balanced:

#color(red)(-1) xx -1x = color(red)(-1) xx 4#

#1x = -4#

#x = -4#