Chapt 6. Rational
Expressions, Functions,
and Equations
6.1 Rational Expressions and
Functions
 Rational


Polynomial divided by non-zero polynomial
120x / (100 – x)
(3x2 - 12xy – 15y2) / (6x3 – 6xy2)
 Rational


Expression
Function
Function defined by a rational expression
f(x) = (120x) / (100 – x)
Evaluating a Function


Given: f(x) = 120x / (100 – x)
Evaluate: f(20)
f(20) = 120(20) / (100 – (20))
= 2400 / 80
= 30
f(40) = 120(40) / (100 – (40))
= 4800 / 60
= 80
Domain of a Rational Function
 Given:
The cost (in $1000) of cleaning up a
polluted lake is a function of the percentage
(x) of the lake’s pollutants to be removed. It
is given by the following function.
f(x) = 120x / (100 – x)
 What
is the cost of cleaning up 50% of the
pollutants?

f(50) = 120(50) / (100 – 50) = 120
Domain of a Rational Function
 Given
the last function:
f(x) = 120x / (100 – x)
 What
are the possible values of x?
 Answer:


x ≠ 100
x cannot be negative (in practical cases)
 Domain

of f:
[0, 2) U (2, 100]
Domain of a Rational Function
f(x) = (2x + 1) / (2x2 – x – 1)
 What is the domain of f?
 Solution:
(2x2 – x – 1)
(2x + 1)(x – 1) = 0
2x + 1 = 0
x–1=0
x = -1/2
x=1
 Given:
Domain of f:
(-∞ , -1/2) U (-1/2, 1) U (1, ∞)
-1/2
1
Your Turn
f(x) = (x – 5) / (2x2 + 5x – 3)
 Find the domain of f.
 Solution:
2x2 + 5x – 3
(2x - 1)(x + 3) = 0
2x – 1 = 0
x+3=0
x=½
x = -3
 Domain of f:
(-∞ , -3) U (-3, 1/2) U (1/2, ∞)
 Given:
Simplifying Rational Expressions
 Simplify:
(x2 + 4x + 3) / (x + 1)
x2 + 4x + 3 (x + 1)(x + 3)
--------------- = ------------------ = x + 1, x ≠ -1
x+1
(x + 1)
y = (x2 + 4x + 3)/(x + 1)
y=x+1
Your Turn

Simplify
1. (x2 + 7x + 10) / (x + 2)

= (x + 2)(x + 5) / (x + 2)
= x + 5, x ≠ -2
2. (x2 – 7x – 18) / (2x2 + 3x – 2)

= (x + 2)(x – 9) / (2x - 1)(x + 2)
= (x – 9) / (2x – 1), x ≠ -2 and x ≠ 1/2
Multiplying Rational Expressions
Multiply




x + 4 x2 – 4x - 21
-------- ∙ ---------------x–7
x2 – 16
x + 4 (x – 7)(x + 3)
= -------- · ------------------x – 7 (x – 4)(x + 4)
x+3
= -------x–4
Dividing Rational Expressions
Divide

(y2 – 25) / (2y – 2)



(y2 + 10y +25) / (y2 + 4y – 5)
= (y2 – 25) / (2y – 2) ∙ (y2 + 4y – 5)/(y2 + 10y + 25)
(y – 5)(y + 5)
(y + 5)(y – 1)
= ------------------ ∙ ------------------2(y – 1)
(y + 5)(y + 5)
y-5
= -------2
Your Turn
Simplify the following


x2 + xy
4x – 4y
----------- · ---------x2 – y2
x


x(x + y)
4(x – y)
= ------------------ · -----------(x – y)(x + y)
x
=4
Your Turn

Simplify

(y2 + y) / (y2 – 4)

= (y2 + y) / (y2 – 4) ∙ (y2 – 1) / (y2 + 5y + 6)


(y2 + 5y + 6) / (y2 – 1)
y(y + 1)
(y + 2)(y + 3)
= ----------------- ∙ -----------------(y – 2)(y + 2) (y - 1)(y + 1)
y(y + 3)
= ------------------(y – 2)(y – 1)
6.2 Adding and Subtracting
Rational Expressions
 Add


x2 + 2x – 2
x + 12
------------------- + -----------------x2 + 3x – 10
x2 + 3x – 10
x2 + 2x – 2 + x + 12
(x + 2) (x + 5)
= ---------------------------- = -------------------x2 + 3x – 10
(x + 5)(x – 2)
(x + 2) (x + 5)
(x + 2)
= -------------------- = ------------(x + 5)(x – 2)
(x – 2)
Your Turn
 Add


x2 – 5x – 15
2x + 5
------------------- + -----------------x2 + 5x + 6
x2 + 5x + 6
Solution
x2 – 5x – 15 + 2x + 5
(x - 5) (x + 3)
= ------------------------------ = -------------------x2 + 5x + 6
(x + 3)(x + 2)
(x + 2) (x + 5)
(x + 2)
= -------------------- = ------------(x + 5)(x – 2)
(x – 2)
Your Turn

Subtract


3y3 – 5x3
4y3 – 6x3
--------------- - --------------x2 – y2
x2 – y2
Solution
3y3 – 5x3 - (4y3 – 6x3)
3y3 – 5x3 - 4y3 + 6x3
= ------------------------------- = ---------------------------x2 – y2
x2 – y2
x3 - y3
(x – y)(x2 + xy + y2) (x2 + xy + y2)
= ---------------- = --------------------------- = -------------------x2 – y2
(x – y)(x + y)
(x + y)
Finding the Least Common
Denominator


Find the LCD of: 7/6x2 & 2/9x
Solution:
1. Factor denominators
6x2  2, 3, x, x
9x  3, 3, x
2. List all factors of 1st Denominator—2, 3, x, x
3. Add factors of 2nd dominator not in the list
—2, 3, x, x, & 3
4. LCD: product of all factors in the list—18x2
Finding the Least Common
Denominator


Find the LCD of:
7/(5x2 + 15x) and 9/(x2 + 6x + 9)
Solution:
1.
2.
3.
4.
Find factors in 1st denominator
5x2 + 15x  5x(x + 3)
Find factors of 2nd denominator
x2 + 6x + 9  (x + 3)(x + 3)
List factors of 1st denominator
5x(x + 3)
Include in the list those factors in 2nd denominator not
found in 1st
5x(x + 3)(x + 3) or 5x(x + 3)2
Your Turn

Find the LCD of:
7 / (y2 – 4) and 15 / (y2 + 2y)
1.
•
•
•
1st den: y2 – 4 = (y + 2)(y – 2)
2nd den: y2 + 2y = y(y + 2)
LCD: (y + 2)(y – 2)y
3/(y2 – 5y – 6) and 6/(y2 – 4y – 5)
2.
•
•
•
1st den: y2 – 5y – 6 = (y – 6)(y + 1)
2nd den: y2 – 4y – 5 = (y – 5)(y + 1)
LCD: (y – 6)(y + 1)(y – 5)
6.3 Complex Rational Expressions

Given:
p =principal (amount borrowed)
 r = monthly interest rate
 n = number of monthly payments
 A = amount of month payment

pr
A = ----------------------1
1 - -------------(1 + r)n
Complex Ration Expression – has complex rational
expression in numerator or denominator


Simplifying
Complex Rational Expression
 Simplify:
1
y
--- + --x
x2
----------1
x
--- + --y
y2
 Find the LCD: x x y y = x2y2
 Multiply all terms by x2y2 / x2y2 = 1

(x2y2)1
(x2y2)y
xy2 + y3
---------- + -------------------------(x2y2)x
(x2y2)x2
x 2y 2
----------------------------- = --------------------(x2y2)1
(x2y2)x
x 2y + x 3
---------- + -------------------------(x2y2)y
(x2y2)y2
x 2y 2
xy2 + y3
y2(x + y)
y2
------------- = -------------- = ----x 2y + x 3
x2(y + x)
x2
Your Turn
1. ((x/y) – 1) / ((x2/y2) – 1))
•
•
Hint: What is the LCD?
Solution: (xy – y2) / (x2 – y2) = y / (x + y)
2. (1/(x + h) – 1/x) / h
•
•
Hint: What is the LCD?
Solution: -1/(x( + h))
Skip
 6.4
Division of Polynomial Expressions
 6.5 Synthetic Division
6.6 Rational Equations
 Given:

Cost (in $1000) of cleaning a lake
120x
f(x) = ---------100 – x
where x = % of pollutants to be eliminated
 Question:

If $80,000 is appropriated for the cleanup,
what % of pollutants can be eliminated?

120x
f(x) = ----------100 – x
 Solution:

200x
80 = ----------100 – x
80(100 – x) = 200x
8000 – 80x = 200x
8000
= 280x
x = 25.7(%)
Solving Rational Equation

Solve:

Note: x ≠ 0
x+6
x + 24
-------- + ---------- = 2
2x
5x
x+6
x + 24
10x -------- + ---------- = 10x 2
2x
5x
5(x + 6) + 2(x + 24) = 20x
5x + 30 + 2x + 48 = 20x
78 = 13x
x=6
Check

Solve:

Note: x ≠ 0
x+6
x + 24
-------- + ---------- = 2
2x
5x
6 + 6 6 + 24 ?
------- + ---------- = 2
2(6)
5(6)
12
30
------- + ------- = 2
12
30
Solving Rational Equation (2)

Solve:
x
3
-------- = ---------- + 9
x–3
x–3
 Note: x ≠ 3
x
3
(x – 3) -------- = (x – 3) --------- + 9
x-3
x-3
x = 3 + (x – 3)9
x = 3 + 9x – 27
x = -24 + 9x
24 = 8x
x=3
But x cannot be 3. Thus, no solution.
Solving Rational Equation (3)

Solve
x
9
---- + ----- = 4
3
x
 Note: x ≠ 0
x
9
(3x) ----- + ---- = (3x) 4
3
x
x(x) + 3(9) = 12x
x2 + 27 = 12x
x2 – 12x + 27 = 0
(x – 3)(x – 9) = 0
x = 3, x = 9
Check

Solve
x
9
---- + ----- = 4
3
x
 Note: x ≠ 0
x = 3, x = 9
3
9?
--- + --- = 4
3
3
9
9 ?
---- + ---- = 4
3
9
1+3=4
3+1=4
Your Turn

Solve:


x+4
x + 20
-------- + ---------- = 3
2x
3x
Solution:

x≠0
x+4
x + 20
6x -------- + ---------- = 6x 3
2x
3x
3(x + 4) + 2(x + 20) = 18x
3x + 12 + 2x + 40 = 18x
52 = 13x
x=4
Your Turn

Solve:


2x
6
-28
-------- + --------- = -----------x–3
x+3
x2 - 9
Solution:

x ≠ 3, x ≠ -3
2x
6
-28
(x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) ----------(x – 3)
(x + 3)
x2 - 9
(x + 3)2x + (x – 3)6 = -28
2x2 + 6x + 6x – 18 = -28
2x2 + 12x + 10 = 0
(2x + 2)(x + 5) = 0
x = -1, x = -5
6.7 Applications

Suppose: Tom can complete a Web site in 15
hours, while her friend Amy can complete it in 10
hours. Working together, how many hours will it
take to complete one job?
 Solution:





Hours working together: x
Hour with Tom alone: 15
Hours with Amy alone: 10
Tom’s rate:
1/15 per hour
Amy’s rate:
1/10 per hour
 Find



an equation
Rate x Time = 1 job
1
1
x ---- + ---- = 1
15
10
(30) x
(30) x
(30) 1
------ ---- + ------ ---- = -----(30) 15 (30) 10
(30)
2x + 3x = 30
5x = 30
x = 6 (hours)
Application
 You
commute to work a distance of 40
miles and return on the same route at the
end of the day. Your average rate on the
return trip is 30 miles per hour faster than
your average rate on the outgoing trip. If
the round trip takes 2 hours, what is your
average rate on the outgoing trip to work

Solution



Average rate outgoing (mph): x
Average rate returning:
x + 30
Find Equation











distance = rate x time
time = distance / rate
(time going) + (time returning) = 2
40/x + 40/(x + 30) = 2
(x + 30)40 + 40x = 2x(x + 30)
40x + 1200 + 40x = 2x2 + 60x
0 = 2x2 - 20x – 1200
0 = x2 - 10x – 600
0 = (x – 30)(x + 20)
x = 30;
x = -20 (has no interpretation)