Why the Sign Changes from < 0 to > 0 ?

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The inequality is a consequence of the fact that the product of two negative numbers is positive and the product of a negative and a positive number is negative. Some textbooks might call it a "basic inequality" or "product rule for inequalities."In summary, this conversation discusses the sign change from negative to positive and the application of the inequality ##ab<0## in different cases. The product rule for inequalities, which states that the product of two negative numbers is positive and the product of a negative and positive number is negative, is also mentioned. It is noted that this is a simple inequality and does not have a specific name.
  • #1
askor
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Why the sign change from < 0 to > 0?
 

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  • #2
This just applies the following:

##a>0,b<0\implies ab<0##
 
  • #3
Math_QED said:
This just applies the following:

##a>0,b<0\implies ab<0##
Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##
 
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  • #4
It would have helped if you had told us what the problem was!

IF the problem was to solve the inequality [itex](1- cos(x))(cos(x)- sin(x))< 0[/itex] then
either
a) [itex]`1- cos(x)> 0[/itex] and [itex]cos(x)- sin(x)> 0[/itex]
or
b)[itex]cos(x)- sin(x)> 0[/itex] and [itex]`1- cos(x)> 0[/itex].
That is because "positive times positive is positive" and "negative time negative is negative".

But you seem to have left out the second case.
 
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  • #5
Thank you for all of your answers but I still don't understand. I even can't find it in my algebra textbook. In what book is this kind of inequality is taught?
 
  • #6
askor said:
Thank you for all of your answers but I still don't understand. I even can't find it in my algebra textbook. In what book is this kind of inequality is taught?
In high school I learned that the product of two negative numbers is positive. And the product of a negative and a positive number is negative.

In general, these inequalities will be in any high school maths textbook or syllabus.
 
  • #7
Why not like this?

##(1 - \text{cos} ~x)(\text{cos} ~x - \text{sin} ~x) < 0##

##(1 - \text{cos} ~x) > 0 ~\vee~ (\text{cos} ~x - \text{sin} ~x) < 0##
 
  • #8
askor said:
Why not like this?

##(1 - \text{cos} ~x)(\text{cos} ~x - \text{sin} ~x) < 0##

##(1 - \text{cos} ~x) > 0 ~\vee~ (\text{cos} ~x - \text{sin} ~x) < 0##
OR allows only one to be true, so ##(1 - \text{cos} ~x) > 0 ~\vee~ (\text{cos} ~x - \text{sin} ~x) < 0## is true in case ##(1 - \text{cos} ~x) > 0 ~\wedge~ (\text{cos} ~x - \text{sin} ~x) > 0##, whereas ##(1 - \text{cos} ~x)(\text{cos} ~x - \text{sin} ~x) < 0## is false.

##ab<0## means exactly one factor is negative while the other one has to be positive. OR doesn't not gurantee the second condition.
 
  • #9
fresh_42 said:
OR allows only one to be true,
No, either or both operands can be true. Possibly you're thinking of "exclusive or" (XOR).
 
  • #10
Mark44 said:
No, either or both operands can be true. Possibly you're thinking of "exclusive or" (XOR).
Yeah, that was a misleading typeset. I didn't meant the Boolean OR, I wanted to emphasize:
<quote> ab < 0 means a>0 and b<0 </quote> or "[ab<0] allows only one to be true"... followed by the explanation that ab<0 is stronger than the Boolean OR. A bit clumsy I admit.
 
  • #11
HallsofIvy said:
either
a) [itex]`1- cos(x)> 0[/itex] and [itex]cos(x)- sin(x)> 0[/itex]
or
b)[itex]cos(x)- sin(x)> 0[/itex] and [itex]`1- cos(x)> 0[/itex].
You didn't mean that. Those two cases are the same.

@askor , The product of two numbers is negative only if exactly one of the numbers is negative and the other is positive.
That gives two cases:
a) 1-cos x < 0 and (cos x - sin x) >0 As in your original attachment
or
b) 1-cos x > 0 and (cos x - sin x) < 0
The first case, (a), which was in your attachment, is not possible because 1-cos x < 0 implies that 1 < cos x. That does not happen for any real number, x.

That only leaves the second case (b) as a possibility.
So we can say that 1-cos > 0 and (cos x - sin x)<0
 
  • #12
fresh_42 said:
Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##

Does above applies too when ab > 0?
 
  • #13
askor said:
Does above applies too when ab > 0?

What are the conditions on ##a## and ##b## if ##ab > 0##?
 
  • #14
PeroK said:
What are the conditions on ##a## and ##b## if ##ab > 0##?

I don't know.
 
  • #15
askor said:
I don't know.

Well, that's an honest answer. Can you think of a way to find out?

You could multiply some numbers together and note whether the product is postive or negative. Then you could look for a pattern in the cases where the product is positive.
 
  • #16
PeroK said:
Well, that's an honest answer. Can you think of a way to find out?

You could multiply some numbers together and note whether the product is postive or negative. Then you could look for a pattern in the cases where the product is positive.

Do you mean if ##ab > 0## then ##(a < 0 \wedge b < 0)## or ##(a > 0 \wedge b > 0)##?
 
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  • #17
askor said:
Do you mean if ##ab > 0## then ##a < 0 \wedge b < 0## or ##a > 0 \wedge b > 0##?

Personally I'd be happy to say that ##ab > 0## if both ##a## and ##b## are positive or both ##a## and ##b## are negative. Which is what you've written in "wedge" notation.
 
  • #18
PeroK said:
Personally I'd be happy to say that ##ab > 0## if both ##a## and ##b## are positive or both ##a## and ##b## are negative. Which is what you've written in "wedge" notation.

Do you mean I was correct?
 
  • #19
askor said:
Do you mean it was correct?

Yes. But, if you don't understand that notation properly I wouldn't use it.
 
  • #20
PeroK said:
Yes. But, if you don't understand that notation properly I wouldn't use it.

The "##\wedge##" notation you mentioned about, it mean "and" isn't it?
 
  • #21
askor said:
The "##\wedge##" notation you mentioned about, it mean "and" isn't it?

Yes. And the other one means "or".
 
  • #22
PeroK said:
Yes. And the other one means "or".

Which one is "or"?
 
  • #23
askor said:
Which one is "or"?

##\vee##

That's your last question.
 
  • #24
What is the name of this inequality?
 
  • #25
askor said:
What is the name of this inequality?
As far as I know, it's too simple to have a name.
 

Related to Why the Sign Changes from < 0 to > 0 ?

1. Why does the sign change from negative to positive?

The sign changes from negative to positive when the value of a number changes from a negative number to a positive number. This is because the sign indicates the direction or orientation of the number on a number line. When the number changes from being on the left side of the number line (negative) to the right side (positive), the sign changes accordingly.

2. What causes the sign to switch from negative to positive?

The sign switches from negative to positive when there is a change in the value of the number. This can happen when performing mathematical operations such as addition, subtraction, multiplication, or division. For example, when adding a negative number to a positive number, the result will be a positive number and the sign will change from negative to positive.

3. Can the sign change from positive to negative?

Yes, the sign can change from positive to negative. This happens when the value of a number changes from a positive number to a negative number. Similar to the previous question, this can occur when performing mathematical operations such as subtraction or division.

4. Is there a specific rule for when the sign changes from negative to positive?

Yes, there is a specific rule for when the sign changes from negative to positive. This rule is based on the properties of numbers and mathematical operations. For example, when multiplying or dividing two numbers with different signs, the result will always be a negative number and the sign will change accordingly.

5. How does the sign change affect the value of the number?

The sign change does not affect the value of the number itself, but it does affect its representation on a number line. The absolute value of the number remains the same, but the sign determines its direction on the number line. For example, the absolute value of -5 and 5 is both 5, but the sign indicates that -5 is on the left side of the number line while 5 is on the right side.

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