MAT 100
Armstrong
Rational Expressions and Equations
Unit 7
Unit Planner
____
7.1
Simplifying Rational Expressions
Read pages 540-546
p. 547 #
____
7.2
Multiplying and Dividing Rational Expressions
Read pages 550-556
p. 556 #
____
7.3
Adding and Subtracting Rational Expressions with Like Denominators; LCD
Read pages 559-565
Day 1 p. 565 #
Day 2 p. 566 #
____
7.4
Adding and Subtracting with Unlike Denominators
Read pages 567-573
p. 573 #
____
7.5
Simplifying Complex Fractions
Read pages 575-581
p. 581 #
____
7.6
Solving Rational Equations
Read pages 584-589
p. 590 #
____
7.7
Problem Solving Using Rational Equations
Read pages 593-596
p. 596 #
____
7.8
Proportions and Similar Triangles
Read pages 599-608
p. 608 #
____
7.9
Variation
Read pages 612-618
p. 618 #
____
Unit 7 Summary/Review
Read pages 622-631
p. 622 #
____
Unit 7 Test
[1]
NOTES:
[2]
Unit 7 – Rational Expressions and Equations
7.1 – Simplifying Rational Expressions
A ______________________________________ is an expression of the form
polynomials and B  0 .
A
where A and B are
B
Evaluating Rational Expressions.
To evaluate rational expressions we will _______________________ and follow order of operations.
Example.
Evaluate
2x  1
x2 1
for x = -3.
Finding Values that Make Rational Expressions Undefined.
To find the values that make a rational expression undefined we will:
1.
_____________________________________________
2.
_____________________________________________
Examples.
Find all real numbers for which each rational expression is undefined:
a.
7x
x 5
b.
[3]
x 1
x2 x 6
Simplifying Rational Expressions.
* To simplify a monomial fraction, we will remove the greatest common factor (GCF) from the
numerator and denominator and cancel.
* To simplify a polynomial fraction, we will:
1. ___________________________________
2. ___________________________________
Examples.
Simplify:
a.
c.
e.
21x 2 y
x 2  3x
b.
3x  9
14xy 2
x 2  13x  12
d.
x 2  144
5x  3   5
7x  3   7
f.
[4]
x3 x2
x 1
x  x  3   3 x  1 
x2 3
Simplifying Rational Expressions that have Opposite Factors.
* If the terms of two polynomials are the same, except they have opposite signs,:
1. _______________________________________
2. _______________________________________
Examples.
Simplify, if possible:
a.
2a  1
1  2a
c.
t 8
t 8
b.
[5]
y 2 1
3  3y
Unit 7 – Rational Expressions and Equations
7.2 – Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions.
Steps:
1. _____________________________________
2. _____________________________________
3. _____________________________________
Examples.
Multiply:
a.
d.
f.
x
2
5
b.
t 1 t 1

t
t 2
e.
3

7 5

9 3x
35x 2 y
7y 2z
c.

x2
2

3
y2
z
5xy
x2 x x 2

2x  4
x
g.
[6]
x 2  3x
x2 x 6

x2 x 2
x2 x
h.
k.
4
x
 3a  1 
j. 5a 

 a 
 1 
i. 63x  

 7x 
x
x2 x
x 2  8 x  7
 x  7 
Dividing Rational Expressions.
Steps:
1. _____________________________________
2. _____________________________________
Examples.
Divide:
a.
a
13

17
26
b. 
9x 15x 2

35 y
14

x 2  x x 2  2x  1

c.
3x  15
6x  30
2 x 2  3x  2
 4x2
d.
2x  1
[7]

Converting Units of Measure.
A roll of carpeting is 12 feet wide and 150 feet long. Find the number of square yards of carpeting
on the roll.
The speed with which light moves through space is about 186,000 miles per second. Express this
speed in miles per hour.
[8]
Unit 7 – Rational Expressions and Equations
7.3 – Adding and Subtracting w/ Like Denominators; LCDs
Adding and Subtracting with Like Denominators.
Steps:
1. ____________________________________________________________________
2. ____________________________________________________________________
3. ____________________________________________________________________
Examples.
Add:
a.
c.
x
8

3x
8
b.
3x  y x  y

5x
5x
3x  21 8x  1

5x  10 5x  10
Subtract:
d.
5x 2 x

3
3
e.
f.
5x  1 4 x  2

x 3 x 3
g.
[9]
5x  1 4 x  2

x 3 x 3
3x  1
x2  x 1

5x  2
x2  x 1

2x  1
x2  x 1
Finding the Least Common Denominator (LCD).
Steps:
1. _____________________________________________________
2. _____________________________________________________
3. _____________________________________________________
Examples.
Find the LCD of each pair of rational expressions.
a.
11
7
and
8x
18x 2
b.
c.
x
x 2
and
7x  7
5x  5
d.
[10]
20
x
and
4x
x 9
6x
2
x  8x  16
and
15x
2
x  16
Building Rational Expression into Equivalent Expressions.
* To build a fraction into an equivalent expression we will always multiply by a form of _________.
Examples.
Write each rational expression as an equivalent expression with the indicated denominator:
a.
c.
7
; denominator 30n 3
15n
x 1
x 2  6x
b.
; denominator x x  6x  2 
[11]
6x
; denominator x  4 x  4 
x 4
Unit 7 – Rational Expressions and Equations
7.4 – Adding and Subtracting w/ Unlike Denominators
Adding and Subtracting Rational Expressions w/ Unlike Denominators.
Steps:
1. ____________________________________________________________________
2. ____________________________________________________________________
3. ____________________________________________________________________
4. ____________________________________________________________________
Examples.
Add or subtract.
a.
c.
d.
4 x 3x

7
5
x 4
x2
x
x 1


b.
5
11

24b 18b
x 5
4x
3
e.
x
[12]
a
a 1

2
a2  1
f.
2a
a 2  4a  4

1
2a  4
g.
4b
b
a5
* When adding or subtracting rational expressions whose denominators are opposite signs, we will
multiply the top and bottom of one fraction by ___________________.
Examples.
a.
3
x

x y y x
b.
[13]
3
2
x y

2
xy

1
xy 2
Unit 7 – Rational Expressions and Equations
7.5 – Simplifying Complex Fractions
A rational expression whose numerator and/or denominator contain fractions is called a
___________________________________.
Simplifying Complex Fractions.
1.
________________________________
2.
________________________________
3.
________________________________
4.
________________________________
Examples.
Simplify:
a.
5x 2
3
2x 3
9
1 1

b. 2 x
x 1

3 5
6
1 1

8 y
d.
8 y
c. x
6
y
y
x
4y 2
[14]
1
e.
1
1
x 1
[15]
Unit 7 – Rational Expressions and Equations
7.6 – Solving Rational Equations
A _____________________________ is an equation that contains one or more rational expressions.
Solving Rational Equations.
Steps:
1. __________________________________________________________________
2. __________________________________________________________________
3. __________________________________________________________________
4. __________________________________________________________________
Examples.
Solve.
a.
c.
x
6

5 1

2 3
b.
22 3a  1 8


5
a
a
d.
[16]
4
x
1 
6
x
x 2
1

1
2
x  3 x  2x  3
e.
4 y  50
4
y 
5
5 y  25
f.
x 3
4

x 1 x 1
Solving for a Variable within a Rational Expression Formula.
The formula
1
r

1
r1

1
r2
is used in electronics to calculate parallel resistances. Solve it for r.
[17]
Unit 7 – Rational Expressions and Equations
7.7 – Problem Solving Using Rational Equations
Solving Number Problems.
If the same number is added to both the numerator and the denominator of the fraction
result is
3
, the
5
4
. Find the number.
5
Solving Shared-Work Problems.
An inlet pipe can fill an oil tank in 7 days, and a second inlet pipe can fill the same tank in 9 days.
If both pipes are used, how long will it take to fill the tank?
[18]
Solving Uniform Motion Problems.
A coach can run 10 miles in the same amount of time as his best student-athlete can run 12 miles.
If the student can run 1 mile per hour faster than the coach, how fast can the student run?
Solving Investment Problems.
At one bank, a sum of money invested for 1 year will earn $96 interest. If invested in bonds, that
money would earn $108, because the interest rate paid by the bonds is 1% greater than that paid by
the bank. Find the bank’s rate.
* Hint: Use the formula
P 
I
rt
where I is the interest, P is the principle (amount invested),
r is the annual rate of interest, and t is the time in years.
[19]
Unit 7 – Rational Expressions and Equations
7.8 – Proportions and Similar Triangles
Writing Ratios and Rates in Simplest Form.
Ratios enable us to compare numerical quantities.



To prepare fuel for a lawnmower, gasoline must be mixed with oil in the ratio of 50 to 1.
To make 14-karat jewelry, gold is mixed with other metals in the ratio of 14 to 10.
In the stock market, winning stocks might outnumber losing stocks in the ratio of 7 to 4.
A __________________ is the quotient of two numbers or the quotient of two quantities that have
the same units.
Examples.
Translate each phrase into a ratio written in fractional form:
a. The ratio 5 to 9.
b.
12 ounces to 2 pounds
A _________________ is a quotient of two quantities that have different units.
A __________________________ is a mathematical statement that two ratios or two rates are equal.
Proportions.
Examples.
Determine whether each equation is a proportion.
a.
3 9

7 21
b.
[20]
8 13

3 5
Solve:
c.
e.
12 3

18 x
a
2

d.
2a  1 10

4
8
4
a2
f.
If 6 apples cost $1.38, how much will 16 apples cost?
g.
A scale is a ratio (or rate) that compares the size of a model, drawing, or map to the size of
an actual object. The scale on a model carousel is 1 inch to that of 160 inches on the actual
carousel. How wide should the model be if the actual carousel is 35 feet wide?
h.
A recipe for rhubarb cake calls for 1
1
1
cups of sugar for every 2 cups of flour. How many
4
2
cups of flour are needed if the baker intends to use 3 cups of sugar?
[21]
Using Proportions to Solve Problems Involving Similar Triangles.
Similar triangles have the same ____________________. In order to have the same shape, the
triangles must have _________________ angle pairs with the same measure. If two triangles are
similar, all pairs of corresponding sides are in ________________________________.
Example.
A tree casts a shadow 18 feet long at the same time as a woman 5 feet tall casts a shadow 1.5 feet
long. Find the height of the tree.
[22]
Unit 7 – Rational Expressions and Equations
7.9 – Variation
If the value of one quantity depends on the value of another quantity, we can often describe that
relationship using the language of variation:



The sales tax on an item varies with the price.
The intensity of light varies with the distance from its source.
The pressure exerted by the water on an object varies with the depth of the object beneath
the surface.
Solving Direct Variation Problems.
* Two variables are said to vary directly if one is a constant multiple of the other.
Two variables that vary directly are represented by the equation:
where k is a constant (number) called the constant of variation.
y  kx
From this formula we can simply calculate the constant of variation, k, using k 
Steps:
1. ___________________________________________
2. ___________________________________________
3. ___________________________________________
4. ___________________________________________
Examples.
Suppose y varies directly as x. If y = 12 when x = 4, find y when x = 6.
[23]
y
.
x
The weight of an object on Earth varies directly with its weight on the moon. If a rock weighs 5
pounds on the moon and 30 pounds on Earth, what would be the weight on Earth of a larger rock
weighing 26 pounds on the moon?
Solving Inverse Variation Problems.
* Two variables are said to vary inversely if one is a constant multiple of the reciprocal of the other.
Two variables that vary inversely are represented by the equation:
y 
where k is a constant (number) called the constant of variation.
k
x
Examples.
Suppose y varies inversely as x. If y = 5 when x = 20, find y when x = 50.
The volume occupied by a gas varies inversely with the pressure placed on it. That is, the volume
decreases as the pressure increases. If a gas occupies a volume of 15 cubic inches when placed
under 4 pounds per square inch (psi) of pressure, how much pressure is needed to compress the
gas into a volume of 10 cubic inches?
[24]