MAT029B
TOPIC: RATIONAL EXPRESSIONS
AND EQUATIONS
6.1 Multiplying and Simplifying Rational Expressions
I.
Rational expressions have an upstairs and a downstairs just like fractions. Please review
fraction facts now (MAT018A) if they’re weak!
II. Something like
is not a number. It is “undefined.” To find these problem spots in rational
expressions, set the bottom (denominator) to zero and solve.
Ex: Find all numbers for which
is not defined.
m2 -25=0
(m+5)(m-5)=0
m= 5,-5
III. Read Directions!
A. “Multiply. Do not simplify.” just means you should insert parentheses and leave alone!
B. “Simplify” means:
1. For monomials Reduce #’s
Subtract exponents
Ex:
2. For Bi-Tri-nomials
Beware First: Factor
Second: Cancel
Ex: Simplify:
=
=
See
a. Factor -1 out of top or bottom
b. Rearrange
c. Cancel
ANS: -1
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C. “Multiply and Simplify” means cross cancel, factoring first if necessary, and never actually
multiply!
Ex:
(t+2)(t-2)
=
t(t-3)
Careful about losing stuff!
6.2 Division an Reciprocals
Just as with the wonderful world of fractions, division involves multiplying by the “reciprocal” or flip.
Ex:
6.3 Least Common Multiples and Denominators
I. Just as with fractions, adding and subtracting are much more complicated than multiplying and
dividing. We need lease common denominators or L.C.D.’s.
II. Let’s begin with least common multiples.
A. Numbers Ex: 8, 36, 40
Step 1: Trees to find prime factorizations
Step 2: Going prime by prime, take the highest number of each prime that shows in any one
factorization.
LCM:
B. Variable monomials Ex:
Step 1: For each variable, find highest exponent
Step 2: LCM:
C. Bi-Tri-nomials Ex:
Step 1: Factor (m-2)(m-3), (m-2)2
Step 2: For each factor find highest exponent
LCM: (m-3) (m-2)2
D. Usually you’ll see combinations of the three forms above. Always factor first and then
consider numbers, variable monomials, and bi-tri-nomials as described above.
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6.4 (Steps for) Adding Rational Expressions
A
Ex:
B
Ex:
C
Ex:
D
Ex:
E
Ex:
F
Ex:
G
Reduced Expression
Ex:
If you want to reverse the order of a binomial denominator such as 5-x, can multiply by -1 as
long as you do the same up top.
6.5 Subtracting Rational Expressions
Same as 6.4 except... be careful subtracting polynomials.
Ex:
6.6 Solving Rational Equations
Trick
I. This is a great trick, but be careful!
Step 1: You must see “=” to use.
Step 2: Create LCD.
Step 3: Don’t use LCD in normal way! Instead write next to each and every term. (Put LCD over
one when dealing with fractions.)
Step 4: Cross cancel.
Step 5: Copy over and finish.
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Step 6: You must check your answer(s) to make sure you never have a zero downstairs.
Previously, checking was a good idea. Here it is the law! Some perfectly good
solutions won’t check!
Ex:
LCD = 5x
(x - 5)(x + 5) = 0
x–5=0
x+5=0
x=5
x = -5
CHECK
?
0=0
?
0=0
6.7 Applications Using Rational Equations and Proportions
I. “Distance = Rate x Time” and “One Complete Job = Rate x Time” word problems
A. Here’s how to organize your work:
1. Make a Box and put all numbers from problem into it. (Look at units!)
RxT=D
RxT=1
A
or
A
1
Book does
B
B
1
this
2. From the box, extract 2 equations in 2 unknowns.
differently!
3. Substitute to get 1 equation in 1 unknown.
4. Clear denominators!
5. Solve and fill in all box numbers to be safe.
B. Tips:
1. If you see “_____-er,” you usually have to add to or subtract from your unknown.
2. “While” means times are both “t”
II. Proportions are very powerful tools: Ratio = Ratio
A. Here’s the setup:
Your placements must be
1.
=
Ex:
consistent. (Look at units!)
2. Cross multiply or clear denominators. (They are the same!)
3. Solve.
B. Tips: Similar Triangles
Lengths of corresponding sides are proportional.
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6.8 Complex Rational Expressions
I. A complex expression is a fractional form with fraction(s) within it. It is not considered
“simplified.”
II. To simplify, pick one of these two methods:
A. Clear the whole mess at once:
1. Find LCD for all internal fractional forms.
2. Don’t use it in the usual way. Write LCD or
next to each and every term.
3. Cross cancel.
4. Clean up.
Ex:
LCD:
B. The honest way:
1. Combine top, combine bottom
2. Flip bottom and multiply.
3. Clean up.
Ex:
(b – a)(b + a)
6.9 Direct and Inverse Variation
I. Suppose you’re riding your bike. You know that the faster you go, the farther you’ll go in the same
time. This is “direct variation”: more speed results in more distance and less speed results in less
distance.
A.
Direct variation can be expressed mathematically as y = k x.
B.
You may notice that this looks a lot like y = mx + b with no b part.
II. Suppose you’re cleaning up after a party. You know that more people working will make the
cleanup take less time. This is “inverse variation’: more people results in less time and fewer
people result in more time.
A. Inverse variation can be expressed mathematically as y = .
B. If you tried to graph this, it would not be a straight line!
III. Steps for solving problems:
A. If you see or understand that the problem involves “direct variation,” write y = kx (or use
more appropriate letters.) Watch your units! Also, order of words in sentence is same as
order of letters in equation. {For “inverse variation” use y = }
B. Find k by plugging in given values and solving.
C. Write out your new customized equation.
D. In fancy problems, you’ll use your new equation.
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