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# Absolute value as distance between numbers

In this video, we think about what |a-b| really means, and we verify that |a-b| = |b-a| by looking at an example.

## Want to join the conversation?

• why 3-(-2) = 5? Could you please give me a example in real world to explain this equation?
• i don't get it how does -5 equal 5
• Absolute value basically measures how far the number is from zero. If you think about a number line, -5 is the same distance away from 0 as 5 is.
Putting an absolute value on something isn't really saying that they are the same number, but it's saying that they are the same distance away from 0.
• Can you help me further understand this concept? I don't really understand how to work it out.
• Every number that are inside the I I becomes positive.
ex)1. I -6 I = I 6 I
2. I -5-2 I = I -7 I = I 7 I
• The practice & exam problems bringing in fractions and decimals is really throwing me off. I looked through the comments, but I don't see any examples that actually address this nor does Sal give any examples.
• With fractions and decimals solving for absolute value is the same process. If I have l 3/4 l and
l -7/6 l being compared then the absolute value of 7/6 is greater.
• Is there a special name for this phenomenon of a formula |a-b| = |b-a|?
• This phenomenon is called commutativity.
So the property |a-b| = |b-a| means that the distance between two numbers is a commutative operation.
• i still don't get how the absolute value of a-b is the same as the absolute value of b-a. Please help.
(1 vote)
• Consider the following:
You have one value, a, that is 3; so a = 3
You also have another, b, that is 7; so b = 7
So a - b = 3 - 7= -4, while b - a = 7 - 3 = 4

When you take the absolute value of either equation (|a - b| or |b - a|), you can see that both result in an answer of 4, as the negative result of a - b (-4) still has a positive absolute value (4).

Let me know if this clarifies, or if you have any further questions!
• Does it always have to be number lines?
Does anyone know of an alternative way to do this?
Also, can anyone properly explain this to me?
Thank You
• No, it does not always have to be on a number line; this is showing you how to do this if you don't have a number line. Since you know you can subtract the number further on the left on the number line from the number on the right to find the distance between, a number line is easy. However, if you have two variables without a number line, you can't know which is greater, so you can't subtract without using this strategy. Since |a-b| = |b-a|, as long as we find the absolute value of the answer, we can subtract in any order.
• this is not really a question but how does he draw straight lines perfectly??
• shouldn't A come before B?
• Only if we are talking about the alphabet. If we are talking about math (algebra) any value can be represented by any letter and that letter can be placed anywhere in the equation. Usually in algebra the letters X and Y are used, but A and B can be used just the same. In this case B has a greater value than A and and A must be subtracted from B
I hope this helps!