How do you solve #8[3-(5+6x)]-4x=-18+2(1-26x)#? Is the equation conditional, an identity, or a contradiction?

First, expand all of the terms in brackets and parenthesis on each side of the equation:

#8color(red)([3 - (5 + 6x)]) - 4x = -18 + color(blue)(2)(1 - 26x)#

#8color(red)([3 - 5 - 6x]) - 4x = -18 + (color(blue)(2) xx 1) - (color(blue)(2) xx 26x)#

#8color(red)([-2 - 6x]) - 4x = -18 + 2 - 52x#

#(8 xx color(red)(-2)) - (8 xx color(red)(6x)) - 4x = -16 - 52x#

#-16 - 48x - 4x = -16 - 52x#

#-16 + (-48 - 4)x = -16 - 52x#

#-16 + (-52)x = -16 - 52x#

#-16 - 52x = -16 - 52x#

Because both sides of the equation are the same, #x# can be anything or, in other words, #x# is the set of all Real Numbers.

Or

#x = {RR}#

• An equation satisfied by every number that is a meaningful replacement for the variable is called an identity.
• An equation satisfied by some numbers but not others, such as 2x =4, is called a conditional equation.
• An equation that has no solution, such as x = x +1, is called a contradiction.

Because this equation is satisfied by every number that is a meaningful replacement, this equation is an identity.