How do you solve #|\frac { 7x - 3} { 5} | = 3#?

Absolute value problems can be confusing. Some teachers say "they make numbers positive," but that's only partially correct. Absolute value functions measure distance. That means, if you stand at point #(0,0)# and move #5# feet, you could either be at point #(5,0)# OR #(-5,0)#. Absolute value measures that, meaning that there are two solutions for each function (as long as there are no domain restrictions). How do you get two solutions? From two equations, of course!

Let's make two equations right now:

#abs((7x-3)/5)=3#

We need to solve for two possibilities, one where the function is negative and one where it's negative:

Positive

#(7x-3)/5=3#

Negative

#-(7x-3)/5=3#

#color(white)(00)#

Now we need to solve both of these:

#(7x-3)/5=3#

multiply by #5# on both sides

#7x-3=15#

add #3# on both sides

#7x=18#

divide by #7#

#x=18/7#

#color(white)(0000)#

Now for the other one:

#-(7x-3)/5=3#

multiply by #-1# on both sides

#(7x-3)/5=-3#

multiply by #5# on both sides

#7x-3=-15#

add #3# on both sides

#7x=-12#

divide by #7# on both sides

#x=-12/7#

Those are our solutions. #x= 18/7,-12/7#

Just to check, let's graph our equation and see if our roots match:

graph{abs((7x-3)/5)-3}

Those are the decimal approximation of our fractions! We're right