Hello, i need help doing a proof, im pretty sure on how to do it but not that way my teacher wants it. He wants it in a chart form by taking splitting up both side.

i have to show that |a-b|=|b-a|

If a>b, it becomes a-b = a-b

If a<b, it becomes b-a = b-a
If a=b it becomes 0 = 0
So it is always true

Sure! I can help you. To prove that |a - b| = |b - a|, we can start with the definition of absolute value. The absolute value of a number x, denoted as |x|, is defined as:

|a - b| =
- (a - b) if a - b < 0
a - b if a - b >= 0

|b - a| =
- (b - a) if b - a < 0
b - a if b - a >= 0

To compare the two expressions, we can split the proof into cases based on the signs of (a - b) and (b - a). There are two cases:

Case 1: (a - b) >= 0 and (b - a) >= 0
In this case, both (a - b) and (b - a) are non-negative. Therefore, |a - b| = a - b and |b - a| = b - a. Since subtraction is commutative, a - b = b - a, which implies that |a - b| = |b - a| in this case.

Case 2: (a - b) < 0 and (b - a) < 0
In this case, both (a - b) and (b - a) are negative. Therefore, |a - b| = -(a - b) and |b - a| = -(b - a). Negating a negative number results in a positive number, so -(a - b) = (b - a). This implies that |a - b| = |b - a| in this case.

Since we have considered all possible cases, we can conclude that |a - b| = |b - a| holds true for any real numbers a and b.

To present this proof in a chart form, you can create a table with two columns: one for the expression |a - b| and another for the expression |b - a|. Then, list the two cases and their corresponding values for each expression, as follows:

|a - b| |b - a|
----------------------------------------------
(a - b) (a - b) if (a - b) >= 0 (b - a)
-(a - b) -(a - b) if (a - b) < 0 -(b - a)

This chart clearly demonstrates the two cases and how both expressions simplify to the same value, proving that |a - b| = |b - a|.

I hope this explanation helps you with your proof! Let me know if you need any further assistance.