How do you solve #7abs(10v-2)-9=5#?

First, add #color(red)(9)# to each side of the equation to isolate the absolute value term while keeping the equation balanced:

#7abs(10v - 2) - 9 + color(red)(9) = 5 + color(red)(9)#

#7abs(10v - 2) - 0 = 14#

#7abs(10v - 2) = 14#

Next, divide each side of the equation by #color(red)(7)# to isolate the absolute value function while keeping the equation balanced:

#(7abs(10v - 2))/color(red)(7) = 14/color(red)(7)#

#(color(red)(cancel(color(black)(7)))abs(10v - 2))/cancel(color(red)(7)) = 2#

#abs(10v - 2) = 2#

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

Solution 1:

#10v - 2 = -2#

#10v - 2 + color(red)(2) = -2 + color(red)(2)#

#10v - 0 = 0#

#10v = 0#

#(10v)/color(red)(10) = 0/color(red)(10)#

#(color(red)(cancel(color(black)(10)))v)/cancel(color(red)(10)) = 0#

#v = 0#

Solution 2:

#10v - 2 = 2#

#10v - 2 + color(red)(2) = 2 + color(red)(2)#

#10v - 0 = 4#

#10v = 4#

#(10v)/color(red)(10) = 4/color(red)(10)#

#(color(red)(cancel(color(black)(10)))v)/cancel(color(red)(10)) = 2/5#

#v = 2/5#

The Solution Set Is: #v = {0, 2/5}#