Algebra Help?

Question

Solve for z:

| 3z - 2 | = | -3z + 3 |

Answer 1

∣3z - 2∣= ∣- 3z + 3∣

remove the absolute value bars

3z - 2 = - 3z + 3

Transpose 3z

3z - 2 + 3z = - 3z + 3 + 3z

Collect like terms

6z - 2 = 3

Transpose 2

6z - 2 + 2 = 3 + 2

Collect like terms

6z = 5

Divide both sides of the equation by 6

6z / 6 = 5 / 6

z = 5/6

- - - - - -- s-

Answer 2

Look at the possibilities. This means writing down what solves the problem. Say you have |A| = |B|, then A = B, -A = B and A = -B are all possibilities that might give a good answer. Just write them out and solve. In two the z terms cancel out and leave something like -2 = 3 which is not a true statement so those can be ignored.

-(3z - 2) = -3z + 3 and -3z =2 = -3z + 3... no answer
3z - 2 = - 3z + 3 and 6z = 5 or z = 5/6
3z - 2 = -(-3z + 3) = 3z + 3 .... no answer

Another way to look at it is to divide by 3.
|z - 2/3| = |-z + 1|

You can then look at each expression as a distance from a point. for example the left expression. If z = 2/3 then it is 0. If z = 1 then it is 1/3 so the left term is how far you are away from the point 2/3.

The left expression is how far you are from the point z=2/3
The right is how far you are from z = 1

Clearly the answer has to be between the two, half way between the two since z (and the distance from 2/3 and 1) has to be the same for both. So:

z = (1/2)( 1 + 2/3) = (1/2)(10/6) = 5/6

Answer 3

| 3z-2| = |-3z+3|
Square both sides to get rid of the absolute brackets because the values in the brackets can either be positive or negative since z is a variable so you don't know whether or not to place a negative to make the value positive.
(|3z-2|)^2 = (|-3z+3|)^2
9z^2-12z+4 = 9z^2-18z+9
-12z+18z = 9-4
6z = 5
z = 5/6