# Find All X Values Satisfying the Equation (Absolute Value)

#### NK8485

##### New member
I'm working on some algebra homework and am a little stumped by this one problem.

It is asking for all x-values that satisfy the equation.

I understand the concept of absolute values, so far the problems have been:

lxl=5 to which I answered: x=-5,5

lx-3l=7 ; x=-4,10.

Now I'm on the problem:
ly-3l=8-ly-3l

I'm not sure if there is a way to do this as a formula or if it's just trial and error of plugging in different numbers to see if they would work out.

#### stapel

##### Super Moderator
Staff member
Now I'm on the problem:
ly-3l=8-ly-3l

I'm not sure if there is a way to do this as a formula or if it's just trial and error...
Notice that the two absolute-value expressions are the same: y - 3. So you've got two cases: y < 3 and y > 3. Consider the cases separately:

. . .y < 3:

. . . . .y - 3 < 0
. . . . .|y - 3| = -(y - 3) = 3 - y
. . . . .8 - |y - 3| = 8 - (3 - y) = 5 + y

Then the equation becomes:

. . . . .3 - y = 5 + y

...and so forth. Do the same thing for the other case. #### lookagain

##### Senior Member
NK8485 & my edit said:
Now I'm on the problem:
ly - 3l = 8 - ly - 3l
Notice that the two expressions inside of the absolute-value bars are the same: y - 3.

$$\displaystyle \ \ \ \$$|y - 3l = 8 $$\displaystyle \$$ - $$\displaystyle \$$ ly - 3l
+ |y - 3| $$\displaystyle \ \ \ \ \ \ \ \$$ + |y - 3|
---------------------------
$$\displaystyle \ \$$2|y - 3| = 8

|y - 3| = 4

Now you can finish it and solve it as you did the others.