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E7 Inequalities - When does it flip?
Contributed by: Cason

  8 > -16

12 > -12

Adding or subtracting any number to both sides of an

inequality does not alter the inequality.  See the proof

below.

-4      -4

16 >  - 8

12 > -12

+4     +4

48 > -48

12 > -12

Multiplying or dividing by a positive also does not alter

the direction of an inequality sign. See proof below.

x4      x4

  3 >  - 3

12 > -12

÷4     ÷4

 -48 < 48

12 > -12

However, multiplying or dividing both sides by a

negative changes the inequalitiy.  See proof below.

•-4   •-4

-3 <  3

12 > -12

÷-4   ÷-4

Did the sign flip?

Undo any multiplication or division with the opposite

operation.  Make sure you use the same sign.

÷3       ÷3
3x  >  12
  x  >  4

Yes

No

Did the sign flip?

Undo any multiplication or division with the opposite

operation.  Make sure you use the same sign.

÷-3   ÷-3
-3x  >  12
  x  <  -4

Yes

No

Did the sign flip?

Undo any multiplication or division with the opposite

operation.  Make sure you use the same sign.

÷3       ÷3
3x  >  -12
  x  > - 4

Yes

No

Did the sign flip?

Undo any multiplication or division with the opposite

operation.  Make sure you use the same sign.

4•
x

4

  x  >  48
   >  12

Yes

No

•4
Did  the sign flip?

Undo any multiplication or division with the opposite

operation.  Make sure you use the same sign.

-4•

-4

x
  x  <  -48
   >  12

Yes

No

•-4

Remember multiplying or

dividing both sides by

a negative flips the

inequality sign.

-4•

-4

x
  x  <  
   >  2
•-4
-5•

-5

x
  x  <  
   >  2
•-5

-4

x
  x  <  
   >  2
x

4

  x  >  
   >  5
  x        12 

6

x
   >  2
>

<

  x        14

-7

x
   >  -2
>

<

  x        -12 

-6

x
   >  2
>

<

  x        16

4

x
   >  4
>

<

  x        75

5

x
   >  15
>

<

  x       -4
3x  >  -12
>

<

  x        6
-2x  >  -12
>

<

  x        -6
3x  >  -18
>

<

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