Chapter 2 IntroduCtIon to VarIables 47

deMYstiFied / algebra deMYstiFied / HuttenMuller / 000-0 / Chapter 2

EXAMPLE

Simplify the fraction:

() 23 () 1

23

+−

+

xx

x

still struggling

When you see a plus or minus sign in a fraction, be very careful when simplify-

ing. For example, we cannot remove x from the numerator and denominator in

the expression^2 +x^ x; x cannot be canceled. the only quantities that can be

divided are factors. Many students mistakenly “cancel” the x and conclude that

221

1 3

+x = + =

x.

to see why we cannot “cancel” x, let us consider a similar fraction that does

not contain a variable.

7

6

43

6

43

6

41

2

5

2

1

2

=^ + =^ + =^ + =

“Canceling” 3 from the numerator and denominator leaves us with the false

equation^76 =^52. the reason that the 3 can’t be canceled from the fraction is that

3 is a term in the numerator, not a factor. (a term is a quantity separated from

others by a plus or minus sign.)

If we wish to divide the numerator and denominator of^2 +x^ x by x we can

rewrite the fraction as the sum of two separate fractions before removing the

common factor.

22 21

1

(^121)
1
+x =+ =+=+
xx
x
xx x^
simply because a plus or minus sign appears in a fraction does not automati-
cally mean that dividing it out is not appropriate. For instance^33 ++xx = 1 because
any nonzero number divided by itself is one.

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EXAMPLE

Simplify the fraction: