Absolute Value Equations
Will someone please check to see if I am doing this correct?
1.|7x+2|=10
x= (8/7) or (-12/7)
2.|3x-5|=-1
I got "no solution" because "d" cannot be negative.
3.|x^2-2x-16|=8
x= 6,-4 or x=4,-2
4.|3x-2|=7
x= 3 or (-5/3)
5.|x^2+2x+1|=-4
I got "no solution" because "d" cannot be negative.
6.|3x+2|+3=0
|3x+2|=-3
Again, I got "no solution" because "d" cannot be negative.
These I don't understand how to do.
7.|3x-1|=|x+4|
8.|(3)/(k-1)|=4
I agree with all your answers, if by "d" you mean the left hand side of the equation.
Regarding no. 7, I suggest you do something like this:
Each side of the equation inside the "absolute bars" could be either positive or negative. Try assuming that each side is one or the other, and try out all four possibilities - like this:
Suppose they're both positive:
3x - 1 = x + 4, so 2x = 5, so x = 5/2. Check it: |15/2 - 1| = |5/2 + 4| Correct.
Suppose the left hand side is positive, but the right hand side is negative:
3x - 1 = -x - 4, so 4x = -3, so x = -3/4. Check it: |-9/4 - 1| = |-3/4 + 4| Correct.
Suppose the left hand side is negative, but the right hand side is positive:
-3x + 1 = x + 4, so -4x = 3, so x = -3/4. But that's just the same as the one above.
Suppose the left hand side and the right hand side are both negative:
-3x + 1 = -x - 4, so 2x = 5, so x = 5/2. But that's just the same as the first one.
So there are two possible answers: x = 5/2 and x = -3/4. (Generally you'll find that you will always get the same two answers from the third and fourth one, because if you negate the entire equation you'll get the earlier solution. It's like saying if -x = -3 then x = 3)
For no. 8, if |3/(k-1)| = 4, then 3/(k-1) equals either +4 or -4. So (k-1)/3 = +4 or -4, so k-1 = +3/4 or -3/4. So k = 1 + 3/4 or 1 - 3/4.