Representation of (x-7) and (x+1)

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Mod note: Changed the title and this post to reflect the situtation in the diagram
the representation of the inequality (x-7)(x+1) ≤ 0 makes no sense to me . can someone explain this .
the diagram is attached
 

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  • #2
BvU
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a times b < 0 if either a or b is negative and the other factor is positive. So they look at that. The vertical lines may be somewhat confusing to you ? The one with the 7 applies to x-7, which changes sign at x =7 and the one with the -1 applies to x = -1.
Both lines apply to the product, that changes sign both at x = -1 and at x = 7. From the +++ and the - - - you can conclude the sign of the product.
 
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  • #3
HallsofIvy
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The top row shows where x- 7 is negative and positive- it is negative for all x< 7, positive for all x> 7.
The second row shows where x- 1 is negative and positive- it is negative for all x< 1, positive for all x> 1.

Since the product of two negative or of two positive number is positive while the product of one negative and one positive number is negative, the diagram shows that the product (x- 7)(x- 1) is positive for x< 1, where both factors, (x- 7) and (x- 1), are negative, positive for x> 7where both factors, (x- 7) and (x- 1), are positive, and negative for 1< x< 7 where (x- 7) is negative and (x- 1) is positive.
 
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  • #4
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The top row shows where x- 7 is negative and positive- it is negative for all x< 7, positive for all x> 7.
The second row shows where x- 1 is negative and positive- it is negative for all x< 1, positive for all x> 1.

Since the product of two negative or of two positive number is positive while the product of one negative and one positive number is negative, the diagram shows that the product (x- 7)(x- 1) is positive for x< 1, where both factors, (x- 7) and (x- 1), are negative, positive for x> 7where both factors, (x- 7) and (x- 1), are positive, and negative for 1< x< 7 where (x- 7) is negative and (x- 1) is positive.
got it Thank you !
 
  • #5
mathman
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the representation of the inequality (x-7)(x-1) ≤ 0 makes no sense to me . can someone explain this .
the diagram is attached
The formula in the diagram is (x-7)(x+1).

Mod note: edited the thread title and post #1 to reflect what's shown in the diagram.
 
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correct ,sorry
 
  • #7
A simple way is that x should be such that product is negative or zero. hence x should be such that only one of two multiples is negative. If number is greater than 7 product will surely be positive and less than -1 both will be negative. I always tell try to understand logically the equations and understanding of euclid's axiom is highly required before using them. Some often gets confused that 0/0 is 1 as 1*0 is 0 and by cross multiplication.....
There's nothing true in cross multiplication we divide both sides by zero and it gives 0/0 is 0/0. Sorry I got away from topic but I thought it would help and makes maths interesting and fun.
 

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