Math III Unit 4: RATIONAL EXPRESSIONS AND
EQUATIONS
Lauren Winstead, Heritage High School
Main topics of instruction:
1) Direct, Joint, and Inverse Variation
2) Rational Functions and their Graphs
3) Multiplying and Dividing Rational Expressions
4) Adding and Subtracting Rational Expressions
5) Solving Rational Equations and Inequalities
Day 1: Direct, Joint, and Inverse Variation
Direct Variation: ______________________________________________________________
_____________________________________________________________________________.
Identifying Direct Variation
Does y vary directly with x?
x
2
3
5
y
8
12
20
x
-1
3
6
y
-2
4
7
What about in equations? Central question: _______________________________________
Example 1:
3y = 2x
You try! Can you get y = 2x + 3 into y = kx form?
Make a table of values, and use it to prove your answer.
x
1
2
3
y
5
7
9
What one rule have you learned from this example? ____________________________________
Example 2: A dripping faucet wastes 1 cup of water if it drips for 3 minutes. The amount of
water wasted varies directly with the amount of time passed. Write an equation of direct
variation.
You try! The circumference of a circle varies directly with the diameter. What is the constant
of variation? Find the diameter of a circle with circumference 105 cm.
Example 3: Write and equation of direct variation that passes through (9, -1).
You try! Write an equation of direct variation that passes through (-3, 14).
Example 4: y varies directly with x, and x = 27 when y = 51. Find x when y = -17.
You try! y varies directly with x. If x = 1 when y = 5, find y when x = 3.
Inverse Variation: _____________________________________________________________
_____________________________________________________________________________.
Modeling Inverse Variation
Example 5: x and y vary inversely. x = 3 when y = -5. Write the function of inverse variation.
You try! Decide which type of variation is represented by the data: direct, inverse, or neither.
x
y
0.5
1.5
2
6
6
18
x
y
0.2
12
0.6
4
1.2
2
x
y
1
2
2
1
3
0.5
Joint Variation
Description
y varies directly with the square of x.
Equation
y varies inversely with the cube of x.
z varies jointly with x and y.
z varies jointly with x and y and inversely with w.
z varies directly with x and inversely with the product of w and y.
Application: The volume of a regular tetrahedron varies directly with the cube of the length of
an edge. The volume of a regular tetrahedron with edge length 3 is
volume of a regular tetrahedron.
9√2
4
. Find the formula for the
Day 2: Rational Functions and Their Graphs
𝑓(𝑥) =
𝑃(𝑥)
𝑄(𝑥)
Points of discontinuity – 2 types
-
Asymptote: _____________________________________________________________
-
Hole: __________________________________________________________________
Example 1: Let’s visualize it! Graph
(𝑥+1)
(𝑥−1)(𝑥+2)
What happened at x = -2 and x = -1?
Try plugging in x = 1 to the equation:
Try plugging in x = 2 to the equation:
The denominator can never equal _____!
Domain:
Example 2: Let’s visualize it! Graph
Range:
(𝑥−2)(𝑥+1)
(𝑥−2)
What happened at x = 2?
Try plugging in x = 2 to the equation:
Why do you think a hole was created
instead of an asymptote?
__________________________________
_________________________________.
Domain:
Range:
Finding Points of Discontinuity
Ask yourself two questions:
1) What numbers will make the denominator equal 0?
2) Do they make a hole or an asymptote?
Example 3: What are the points of discontinuity of 𝑦 =
1
?
𝑥 2 +2𝑥+1
Is it an asymptote or a hole? Graph it!
You try! Find the points of discontinuity for:
a) 𝑦 =
1
𝑥 2 −16
b) 𝑦 =
𝑥+1
𝑥 2 +3
You try! Find the points of discontinuity for 𝑦 =
(𝑥+3)(𝑥−2)
. Are they holes or asymptotes?
(𝑥−2)(𝑥+1)
Day 3: More Rational Function Graphs: Horizontal Asymptotes
Yesterday, we talked about _______________ asymptotes and _____________.
But, some graphs also have _______________ asymptotes.
Rules for Horizontal Asymptotes
1. Top Heavy: There is a horizontal asymptote at ____________ if ___________________
__________________________________________________________________________.
2. Bottom Heavy: There is a horizontal asymptote at ____________ if ________________
__________________________________________________________________________.
3. Balanced: There is a horizontal asymptote at ____________ if ___________________
__________________________________________________________________________.
Example 1: Where are the points of discontinuity for 𝑦 =
and/or holes?
𝑥+2
, including all asymptotes
2𝑥 2 −4
You try! What are all the points of discontinuity?
a) 𝑦 =
b) 𝑦 =
𝑥 2 +4
Domain:
𝑥−1
Domain:
Range:
Domain:
Range:
Range:
(𝑥−2)(𝑥+3)
(𝑥+3)(𝑥−1)
Day 4: Multiplying and Dividing Rational Expressions
There is one goal when it comes to multiplying and dividing rational expressions, and that is to
____________________________________________________________________________.
Example 1: Simplify
𝑥 2 +10𝑥+25
𝑥 2 +9𝑥+20
What are the restrictions on x? (What would make the denominator = 0?) __________________
You try! Simplify: a)
27𝑥 3 𝑦
b)
9𝑥 4 𝑦
−6−3𝑥
𝑥 2 −6𝑥+8
Restrictions: ____________________
Restrictions: _______________________
Domain: ________ Range: ________
Domain: ___________ Range: __________
Multiplying Rational Expressions
Example 2: Multiply
2𝑥 2 +7𝑥+3
𝑥−4
∙
𝑥 2 −16
𝑥 2 +8𝑥+15
Restrictions: ____________________________________ Domain: ________ Range: ________
You try! Multiply. a)
𝑥+7
7𝑥+35
∙
𝑥 2 −3𝑥−40
𝑥−8
b)
45𝑥 2
𝑥−9
∙
𝑥 2 −5𝑥−36
3𝑥 3 +12𝑥 2
Restrictions: ___________________
Restrictions: ___________________
Domain: ________ Range: _________
Domain: _______ Range: ________
Dividing Rational Expressions
Example 3: Divide
4−𝑥
(3𝑥+2)(𝑥−2)
÷
5(𝑥−4)
𝑥−2)(7𝑦−5)
Restrictions: _____________ Domain: ______ Range: ______
What should we always
remember when dividing
fractions?
When should you decide your
restrictions?
__________________________.
Example 4: Divide
7𝑎2
7𝑎3 +56𝑎2
÷
2
𝑎2 +7𝑎−8
Restrictions: _________________ Domain: ____________ Range: _____________
You try! Divide a)
𝑏2 −2𝑏−15
8𝑏+20
÷
2
4𝑏+10
Restrictions: _______________________
Domain: __________ Range: _________
b)
10𝑥 2 −28𝑥+16
2𝑥−4
÷
25𝑥 2 −25𝑥+4
5𝑥 2 −41𝑥+8
Restrictions: ___________________
Domain: ________ Range: _______
Day 5: Adding and Subtracting Rational Expressions
Just like addition and subtraction with all fractions, the main thing to focus on is _____________
_____________________________________________________________________________.
Adding Rational Expressions
Example 1: Add
3
𝑥+6
+
7
𝑥−2
Ask yourself how you can make the denominators the same!
What are the restrictions on x? _______________________________
You try! Add
7𝑛
𝑛+1
+
8
𝑛−7
What are the restrictions on x? ___________________________
Example 2: Add
1
𝑥 2 +5𝑥+4
+
5𝑥
.
3𝑥+3
Factor first and ask yourself how you can make the denominators the same!
What are the restrictions on x? ___________________________
You try! Add
5𝑛+5
5𝑛2 +35𝑛−40
7
+ 3𝑛
What are the restrictions on x? ______________________________
7
Example 3: Subtract
3
−
8
.
12𝑥−8
What are the restrictions on x? ___________________________
You try!
5
𝑛+5
−
4𝑛
2𝑛+6
What are the restrictions on x? ___________________________
Example 4: Subtract
5𝑥−4
2𝑥 2 +3𝑥−9
−
𝑥+6
5𝑥 2 +19𝑥+12
What are the restrictions on x? ___________________________
You try! Subtract
7𝑦
5𝑦 2 −125
−
4
3𝑦+15
What are the restrictions on x? ___________________________
Day 6: Complex Rational Expressions
Simplifying complex rational expressions involves ____________________________________.
Example 1: Simplify
𝑥−2
2
+
𝑥
𝑥+1
3
1
−
𝑥−1 𝑥+1
First simplify the top:
Then, simplify the bottom:
Then, keep/change/flip:
What are the restrictions on x? ________________________________
You try! Simplify
16
4
−
𝑚−3 𝑚−4
16 𝑚−4
−
𝑚2 𝑚−3
Try one more! Simplify
1
+𝑥+3
𝑥−1
1
𝑥−3+𝑥+4
Day 7: Solving Rational Equations and Inequalities
This time, you won’t just be simplifying, you’ll be solving!
Example 1: What are the solutions of the rational equation?
Example 2: Find the solutions!
You try! Find the solutions of
𝑥
𝑥−3
𝑥
𝑥+1
+
+
𝑥
𝑥+3
3
𝑥+4
=
=
2
𝑥 2 −9
𝑥+3
𝑥+4
5
𝑥−2
=
15
𝑥 2 −1
Example 3: Find the solutions!
You try! Find the solutions!
𝑥−1
𝑥 2 +3𝑥+2
𝑥−1
𝑥+2
=
+
2𝑥
𝑥+2
=
𝑥−1
𝑥+1
𝑥 2 +2𝑥−3
𝑥+2
Rational Inequalities
Example 1: Find the solutions for
You try! Find the solutions for a)
𝑥
4−𝑥
2
𝑥−1
< 3.
<𝑥
b) 𝑥 + 1 >
𝑥+5
𝑥+2
Day 8: Applications of Rational Equations
Example 1: A flight across the United States takes longer east to west than it does west to east.
Assume that winds are constant in the eastward direction. When flying westward, the headwind
decreases the airplane’s speed. When flying eastward, the tailwind increases its speed. The time
for a round trip is 7.75 hours. If the airplane cruises at 480 mi/hr, what is the speed of the wind?
Step 1: What is your basic equation for distance, rate, and time?
Distance
Rate
Time
Going West to East
Going East to West
Total time = _____________________ + __________________________
You try! You ride your bike to a store 4 miles away to pick up things for dinner. When there is
no wind, you ride at 10 mph. Today, your trip to the store and back took 1 hour. What was the
speed of the wind today?
Example 2: You can stuff envelopes twice as fast as your friend. Together, you can stuff 6750
envelopes in 4 hours. How long would it take each of you working alone to complete the job?
You try! On the first four tests of the term, your average is 84%. You think you can score 96%
on each of the remaining tests. How many consecutive test scores of 96% would you need to
bring your average up to 90% for the term?