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Removing grouping symbols:  Level 2

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 The relationship of  ab  to  ba

 


We know the relationship of  a + b  to  b + a:

a + b = b + a.

But what is the relationship of  ab  to  ba ?

ab is the negative of ba.

ab  =  −(ba).

For example, 2 − 5 = −(5 − 2).

Problem 20.   Prove:   ab = −(ba).

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  Solution 1. ab = a + (−b)
 
  = b + a
 
  = −(ba).

Solution 2.   Two numbers are negatives of one another if their sum is 0.

If  
  p + q  =  0,
 
then  
 
  p  =  q.

ab  and  ba  clearly satisfy that test:

ab  +  ba = 0.

Therefore, ab = −(ba).

   Problem 21.    x − 2
2 − x
 =  −1
The fraction has the form:     a 
a
 , which for all a (except 0) is equal

to −1.

The rule:   a
b
  =   a
b

We may interpret that rule (Lesson 4) to mean:  

In any fraction we may change the signs
in both the numerator and denominator.

  Example.   Apply that rule to   pq
  −2
.
  Answer.     pq
  −2
qp
   2

qp is the negative of pq.

Problem 22.   Apply that rule.

  a)   ab
 −c
 =  ba
  c
  b)   2 − x
 −5
 =  x − 2
   5
  c)   x + yz
     −4
 =  xy + z
     4
    d)   x − 1
    2
 =  x + 1
 −2
 =  x + 1
   2

Next Lesson:  Adding like terms


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