Math 96–Algebraic Fractions (Rational Expressions)–page 1
Reduce. The concept of reducing rational expressions is to factor to reduce. In fact, if you can’t factor, you
can’t reduce! Notice, when you reduce, you are using a “form of one”.
a.
b.
c.
Notice on example c that the negative symbol in the denominator was “moved” to the front!
Multiply. The concept is to factor to reduce. Again, you reduce by using a “form of one”.
d.
Divide. The concept is to invert the divisor, factor, reduce. You reduce by using a “form of one”.
e.
Add/Subtract. The concept is to get a common denominator (you may have to factor the denominator first)
so multiply each fraction by a form of 1 to obtain the common denominator, KEEP the common
denominator, distribute and combine the numerators, check for factoring and reducing at the end of
the problem.
f.
g.
h.
Math 96–Algebraic Fractions (Rational Expressions)–page 2
Solve. The concept is to multiply the entire equation by one common denominator; this will eliminate all
denominators. Then you solve the resulting equation. When the original denominator has variables,
there will be some numbers that cannot be used for the solution.
i.
Multiply by LCD of 30
6(2x + 7) ! 15(3x ! 4) = 5(4x + 2)
12x + 42 ! 45x + 60 = 20x + 10
! 33x + 102 = 20x + 10
! 53x + 102 = 10
! 53x = ! 92
or
j.
Multiply by LCD of (x+ 3)(x! 3)
x cannot be ! 3 or 3 so x … ! 3, 3
4(x ! 3) + 7 = 5(x + 3)
4x ! 12 + 7 = 5x + 15
4x ! 5 = 5x + 15
! 1x ! 5 = 15
! 1x = 20
x = ! 20 (can you keep this answer? yes!) or
{! 20}
k.
Multiply by LCD of (x+ 6)(x! 6); x … ! 6, 6
24 + 8(x ! 6) = 2(x + 6)
24 + 8x ! 48 = 2x + 12
8x ! 24 = 2x + 12
6x ! 24 = 12
6x = 36
x = 6 (can you keep this answer? no! this is one of the numbers that can’t be used!)
empty set or i
Math 96–Algebraic Fractions (Rational Expressions)–page 3
Complex Fractions. The concept is to obtain a result with only one numerator and one denominator. There
are two approaches. One approach is to get a common denominator within the numerator and a
common denominator within the denominator. Then it’s numerator divided by denominator: invert
the divisor (the denominator); factor and reduce if possible; see example a. The other approach is to
multiply all terms by the same common denominator to simplify all terms; then factor and reduce if
possible; see example b. Some people think method b is quicker than method a.
l.
Notice that you are multiplying each individual fraction by a form of 1 so you are changing
how the fraction looks, not changing its value.
m.
Notice that you get the same answer with each approach shown above. You just need to
decide which way makes the most sense to you and practice that method. Also, notice on
example b, you are multiplying the original fraction by a form of 1 so you are changing how
the fraction looks, not changing its value.
Math 96–Algebraic Fractions (Rational Expressions)–page 4
Evaluate. Observe the following and try to figure out the logic behind each step.
n.
Evaluate
OR
Notice you get the same result!