Absolute Value

[sologuitar]

The absolute value of a real number a, denoted by the symbol | a |, is defined by the rules below.

Rule 1

| a | = a if a is >= 0

Rule 2

| a | = - a if a < 0.

Rule 2 is a bit unclear. Can someone provide 2 examples to clarify Rule 2?

 

[Phrzby Phil]

What is unclear about it?

 

[sologuitar]

What is unclear about it?

Why is the | a | = - a?

 

[e.jane.aran]

The absolute value of a real number a...

Rule 2

| a | = - a if a < 0.

Rule 2 is a bit unclear. Can someone provide 2 examples to clarify Rule 2?

I suspect the confusion is down to absolute values always being positive (or at least non-negative), and "minus" signs "always" indicating negatives. But this misunderstands the "minus" sign. (You're not at all alone in this confusion, by the way.)

The "minus" sign means only and exactly "the opposite of the original sign". So the opposite of +2 is -2, but the opposite of -2 is -(-2) = +2.

In the context of absolute values, if "a" is negative, then its absolute value is positive, which means that the sign on "a" is now the opposite. So:

|-2| = -(-2) = +2

That's all the "-" means here.

 

[sologuitar]

I suspect the confusion is down to absolute values always being positive (or at least non-negative), and "minus" signs "always" indicating negatives. But this misunderstands the "minus" sign. (You're not at all alone in this confusion, by the way.)

The "minus" sign means only and exactly "the opposite of the original sign". So the opposite of +2 is -2, but the opposite of -2 is -(-2) = +2.

In the context of absolute values, if "a" is negative, then its absolute value is positive, which means that the sign on "a" is now the opposite. So:

|-2| = -(-2) = +2

That's all the "-" means here.

I totally get it now. Thanks.

Three extra examples.

| - 30 | = - (-30) = 30

| -1/2 | = -(-1/2) = 1/2

Absolute value is distance, right? I know that distance is positive.

What about an expression inside the absolute value?

| -(2x + 4) |

-[-( 2x + 4)]

-(-2x - 4)

2x + 4

Yes?

 

[Phrzby Phil]

I see the above replyers have cleared it up for you.

Let me add further, this is what is known as a piecewise function, meaning that the expressions to evaluate it differ for different parts (points or intervals) of the function's domain.

For |x|, we consider 2 parts of the domain: <0 and >= 0.

You can see that for the former, "-a" is the function's value; for the latter, just "a" .

 

[sologuitar]

I see the above replyers have cleared it up for you.

Let me add further, this is what is known as a piecewise function, meaning that the expressions to evaluate it differ for different parts (points or intervals) of the function's domain.

For |x|, we consider 2 parts of the domain: <0 and >= 0.

You can see that for the former, "-a" is the function's value; for the latter, just "a" .

You said:

"Let me add further, this is what is known as a piecewise function, meaning that the expressions to evaluate it differ for different parts (points or intervals) of the function's domain."

Can you show me what you mean using a piecewise function?

 

[Phrzby Phil]

Typically a book will use a big curly bracket (which I cannot do here) to identify the expression for each piece of the domain.

So defn of abs value is:
|x| = { -x, if x < 0
{ x, if x >= 0

Here's another - a step function, because the graph looks like a set of steps:
[[x]] = the greatest integer less than or equal to x.
Examples:
[[-2]] = -2
[[-1.5]] = -2
[[0]] = 0
[[0.5]] = 0
[[1]] = 1
[[1.5]] = 1

 

[sologuitar]

Typically a book will use a big curly bracket (which I cannot do here) to identify the expression for each piece of the domain.

So defn of abs value is:
|x| = { -x, if x < 0
{ x, if x >= 0

Here's another - a step function, because the graph looks like a set of steps:
[[x]] = the greatest integer less than or equal to x.
Examples:
[[-2]] = -2
[[-1.5]] = -2
[[0]] = 0
[[0.5]] = 0
[[1]] = 1
[[1.5]] = 1

I thank you. You can draw graphs using desmos and express any function by hand to upload here. I do it all the time. I am not too familiar with piecewise functions and step functions. Perhaps, I will post some questions concerning both when time allows.

 

[solo_guitar]

What is unclear about it?

I hope you are doing great. I am considering a chapter by chapter, section by section review of College Algebra. The textbook I will be using is College Algebra by Michael Sullivan Edition 9, which you can free download by simply searching online.

Come along with me as I cover every chapter, every section in the Michael Sullivan Edition 9 textbook. It's going to be fun. We are going to learn so much from each other. Knowledge is power.

My name is Feliz. What is yours? So, are you going to take this ride with me through college algebra on our way to precalculus and finally calculus l, ll, and lll? You say?

Feliz

 

[solo_guitar]

I suspect the confusion is down to absolute values always being positive (or at least non-negative), and "minus" signs "always" indicating negatives. But this misunderstands the "minus" sign. (You're not at all alone in this confusion, by the way.)

The "minus" sign means only and exactly "the opposite of the original sign". So the opposite of +2 is -2, but the opposite of -2 is -(-2) = +2.

In the context of absolute values, if "a" is negative, then its absolute value is positive, which means that the sign on "a" is now the opposite. So:

|-2| = -(-2) = +2

That's all the "-" means here.

I hope you are doing great. I am considering a chapter by chapter, section by section review of College Algebra. The textbook I will be using is College Algebra by Michael Sullivan Edition 9, which you can free download by simply searching online.

Come along with me as I cover every chapter, every section in the Michael Sullivan Edition 9 textbook. It's going to be fun. We are going to learn so much from each other. Knowledge is power.

My name is Feliz. What is yours? So, are you going to take this ride with me through college algebra on our way to precalculus and finally calculus l, ll, and lll? You say?

Feliz

 

[conway]

Distance....proves zero has dimension....therefore....it has a solution to division!

 

[solo_guitar]

Distance....proves zero has dimension....therefore....it has a solution to division!

I am strictly interested in math solutions and steps. I am not interested in philosophy. Understand?

 

[conway]

All the original mathematicians were philosophers! Try me on this truth...I am ready to copy and paste well knowen facts!