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Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc.
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GLENCOE DIVISIONGlencoe/McGraw-Hill8787 Orion PlaceColumbus, Ohio 43240
Lesson 12-1 Inverse Variation
Lesson 12-2 Rational Expressions
Lesson 12-3 Multiplying Rational Expressions
Lesson 12-4 Dividing Rational Expressions
Lesson 12-5 Dividing Polynomials
Lesson 12-6Rational Expressions with Like Denominators
Lesson 12-7Rational Expressions with Unlike Denominators
Lesson 12-8Mixed Expressions and Complex Fractions
Lesson 12-9 Solving Rational Equations
Example 1 Graph an Inverse Variation
Example 2 Graph an Inverse Variation
Example 3 Solve for x
Example 4 Solve for y
Example 5 Use Inverse Variation to Solve a Problem
Manufacturing The owner of Superfast Computer Company has calculated that the time t in hours that it takes to build a particular model of computer varies inversely with the number of people p working on the computer. The equation can be used to represent the people building a computer. Complete the table and draw a graph of the relation.
p 2 4 6 8 10 12
t
p 2 4 6 8 10 12
t
Original equation
Replace p with 2.
Divide each side by 2.
Simplify.
6
Solve the equation for the other values of p.
3 2 1.5 1.2 1
Answer:
Solve for
Answer:
Graph the ordered pairs: (2, 6), (4, 3), (6, 2), (8, 1.5), (10, 1.2), and (12, 1).
As the number of people p increases, the time t ittakes to build a computer decreases.
p 2 4 6 8 10 12
t
p 2 4 6 8 10 12
t 18 9 6 4.5 3.6 3
Answer:
Manufacturing The foreman of a package delivery company has found that the time t in hours that it takes to prepare packages for delivery varies inversely with the number of people p that are preparing them. The equation can be used to represent the people preparing the packages. Complete the table and draw a graph of the relation.
Answer:
Inverse variation equation
The constant of variation is 4.
Graph an inverse variation in which y varies inversely as x and
Solve for k.
x y
–4 –1
–2 –2
–1 –4
0 undefined
1 4
2 2
4 1
Choose values for x and y whose product is 4.
Answer:
Graph an inverse variation in which y varies inversely as x and
Method 1 Use the product rule.
Product rule for inverse variations
Divide each side by 15.
Simplify.
If y varies inversely as x andfind x when
Method 2 Use a proportion.
Proportion rule for inverse variations
Cross multiply.
Divide each side by 15.
Answer: Both methods show that
If y varies inversely as x andfind x when
Answer: 8
If y varies inversely as x and find y when
Use the product rule.
Product rule for inverse variations
Divide each side by 4.
Simplify.
Answer:
If y varies inversely as x and find y when
Answer: –25
Physical Science When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. How far should a 2-kilogram weight be from the fulcrum if a 6-kilogram weight is 3.2 meters from the fulcrum?
Original equation
Divide each side by 2.
Simplify.
Answer: The 2-kilogram weight should be 9.6 meters from the fulcrum.
Answer: 1 m
Physical Science How far should a 10-kilogram weight be from the fulcrum if a 4 kilogram weight is 2.5 meters from the fulcrum?
Example 1 One Excluded Value
Example 2 Multiple Excluded Values
Example 3 Use Rational Expressions
Example 4 Expression Involving Monomials
Example 5 Expression Involving Polynomials
Example 6 Excluded Values
Exclude the values for which
Subtract 7 from each side.
Answer: b cannot equal –7.
The denominator cannot equal zero.
State the excluded value of
Answer: –3
State the excluded value of
Exclude the values for which
The denominator cannot equal zero.
Factor.
Use the Zero Product Property to solve for a.
or
Answer: a cannot equal –3 or 4.
State the excluded values of
Answer: 2, 3
State the excluded values of
The original mechanical advantage was 5.
Landscaping Refer toExample 3 on page 649.Suppose Kenyi finds arock that he cannot movewith a 6-foot bar, so he gets an 8-foot bar. But thistime, he places the fulcrumso that the effort arm is 6 feetlong, and the resistance armin 2 feet long.
Explain whether he has more or less mechanicaladvantage with his new setup.
Simplify.
Answer: Even though the bar is longer, because he moved the fulcrum he has a mechanical advantage of 3, so his mechanical advantage is less than before.
Use the expression for mechanical advantage to write an expression for the mechanical advantage in the new situation.
Answer: Since the mechanical advantage is 3, Kenyi canlift 3 • 180 or 540 pounds with the longer bar.
If Kenyi can apply a force of 180 pounds, what is the greatest weight he can lift with the longer bar?
Landscaping Sean and Travis are responsible for clearing an area for a garden. They come across a large rock that they cannot lift. Therefore, they use a 5-foot bar as a lever, and the fulcrum is 1 foot away from the rock.
a. Use the formula to find the mechanical advantage.
b. If they can apply a force of 200 pounds, what is the greatest weight they can lift?
Answer: 4
Answer: 800 lb
The GCF of the numeratorand denominator is
Divide the numerator anddenominator by
1
1
Simplify
Answer: Simplify.
Simplify
Answer:
Factor.
Divide the numerator and denominator by the GCF, x – 7.
1
1
Simplify
Answer: Simplify
Simplify
Answer:
Divide the numeratorand denominator bythe GFC, x + 4.
1
1
Simplify State the excluded values of x.
Factor.
Simplify.Answer:
Exclude the values for which equals 0.
The denominator cannot equal zero.
Factor.
Zero Product Property
Evaluate.
Simplify.
Check Verify the excluded values by substituting them into the original expression.
Evaluate.
Simplify.
Answer: The expression is undefined when andTherefore,
Answer:
Simplify State the excluded values of w.
Example 1 Expressions Involving Monomials
Example 2 Expressions Involving Polynomials
Example 3 Dimensional Analysis
Method 1 Divide by the greatest common factor after multiplying.
Multiply the numerators.
Multiply the denominators.
The GCF is 98xyz.
Simplify.
Find
Method 2 Divide the common factors before multiplying.
Multiply.Answer:
Divide by common factors and z.
1 x 1
6 z 2 y
31 1
1 1
Multiply.Answer:
Divide bycommon factors and r.
1 1 1
1 1 2 1 1
d 2 q
2 r 3
Find
Answer:
Answer:
a. Find
b. Find
Factor thenumerator.
Simplify.Answer:
The GCF is1
1
1
x 2
Find
Find
Factor.
The GCF is1
1
1
1
Multiply.
Simplify.Answer:
a. Find
b. Find
Answer:
Answer:
Space The velocity that a spacecraft must have in order to escape Earth’s gravitational pull is called the escape velocity. The escape velocity for a spacecraft leaving Earth is about 40,320 kilometers per hour. What is this speed in meters per second?
Answer: The escape velocity is 11,200 meters per second.
1120 10
1 1
Simplify.
Multiply.
Example 3-3b
Aviation The speed of sound, or Mach 1, is approximately 330 meters per second at sea level. What is the speed of sound in kilometers per hour?
Answer: 1188 kilometers per hour
meters
km
hour
meters
1000
sec3600
sec
330
Example 1 Expression Involving Monomials
Example 2 Expression Involving Binomials
Example 3 Divide by a Binomial
Example 4 Expression Involving Polynomials
Example 5 Dimensional Analysis
Find
Multiply by
the reciprocal of
Answer: Simplify.
Divide by common factors 5, 6, and x.
1
4 11
15x3
Find
Answer:
Find
Multiply by
the reciprocal
of
Factor
or Simplify.Answer:
The GCFis
1
1
Find
Answer:
Find
Multiply by
the reciprocal
of
Factor
Simplify.Answer:
The GCF is
1
1
Find
Answer:
Find
Multiply by the
reciprocal,
Factor
Simplify.Answer:
The GCFis
1
1
Find
Answer: or (k – 10) (k – 2)
10
Aviation In 1986, an experimental aircraft named Voyager was piloted by Jenna Yeager and Dick Rutan around the world non-stop, without refueling. The trip took exactly 9 days and covered a distance of 25,012 miles. What was the speed of the aircraft in miles per hour? Round to the nearest mile per hour.
Use the formula for rate, time, and distance.
Divide each side by 9 days.
Convert days to hours.
Answer: Thus, the speed of the aircraft was about 116 miles per hour.
Aviation Suppose that Jenna Yeager and Dick Rutan wanted to complete the trip in exactly 7 days. What would be their average speed in miles per hour for the 25,012-mile trip?
Answer: about 149 miles per hour
Example 1 Divide a Binomial by a Monomial
Example 2 Divide a Polynomial by a Monomial
Example 3 Divide a Polynomial by a Binomial
Example 4 Long Division
Example 5 Polynomial with Missing Terms
Find
Write as a rational expression.
Divide each term by 2x.
2x 9
11
Simplify each term.
Simplify.Answer:
Find
Answer:
Find
Write as a rational expression.
Divide each term by 3y.
Simplify.Answer:
2y 1
13
Simplify each term.
y
3
Find
Answer:
1
1
Divide by the GCF.
Find
Factor the numerator.
Simplify.Answer:
Write as a rational expression.
Find
Answer:
Find
Step 1 Divide the first term of the dividend, x2, by the first term of the divisor, x.
x
Multiply x and x – 2.
Subtract.
Step 2 Divide the first term of the partial dividend, 9x – 15, by the first term of the divisor, x.
x + 9
Subtract and bring down –15.
Multiply 9 and x – 2.
Subtract.
Answer: The quotient of is
with a remainder of 3, which can be written as
Find
Answer: The quotient is with a remainder of 2.
Rename the x2 term by using a coefficient of 0.
Find
Multiply x2 and x – 5.
Subtract and bring down –34x.
Multiply 5x and x – 5.
Subtract and bring down 45.
Multiply –9 and x – 5.
Subtract.
Answer:
Find
Answer: The quotient is
Example 1 Numbers in Denominator
Example 2 Binomials in Denominator
Example 3 Find a Perimeter
Example 4 Subtract Rational Expressions
Example 5 Inverse Denominators
Find
The common denominator is 15.
Add the numerators.
Simplify.Answer:
Divide by the common factor, 5.
4
3
Find
Answer:
Divide by the common factor, c + 2.
1
1
Find
The common denominator is c + 2.
Factor the numerator.
Simplify.Answer:
Answer: 5
Find
Geometry Find an expression for the perimeter of rectangle WXYZ.
Perimeter formula
The commondenominator is
Distributive Property
Combine like terms.
Factor.
Answer: The perimeter can be represented by the
expression
Geometry Find an expression for the perimeter of rectangle PQRS.
Answer:18
Find
The common denominator is
The additive inverse ofis
Distributive Property
Simplify.Answer:
Find
Answer:
Find
The denominator is the same asor . Rewrite the second expression so that it has the same denominator as the first.
Rewrite usingcommon denominators.
The common denominator is
Simplify.Answer:
Find
Answer:
Example 1 LCM of Monomials
Example 2 LCM of Polynomials
Example 3 Monomial Denominators
Example 4 Polynomial Denominators
Example 5 Binomials in Denominators
Example 6 Polynomials in Denominators
Find the LCM of
Find the prime factors of each coefficient and variable expression.
Use each prime factor the greatest number of times it appears in any of the factorizations.
Answer:
Find the LCM of
Answer:
Find the LCM of
Express each polynomial in factored form.
Use each factor the greatest number of times it appears.
Answer:
Find the LCM of
Answer:
Find
Factor each denominator and find the LCD.
Since the denominator of is already 5z, onlyneeds to be renamed.
Multiply
by
Distributive Property
Add the numerators.
Answer: Simplify.
Find
Answer:
Find
Factor thedenominators.
The LCDis
Add thenumerators.
Simplify.Answer:
Find
Answer:
Find
Factor.
The LCD is
Add thenumerators.
Multiply.
Simplify.Answer:
Find
Answer:
Multiple-Choice Test Item
Find
A B
C D
Read the Test Item
The expression represents
the difference of two rational expressions with
unlike denominators.
Solve the Test ItemStep 1 Factor each denominator and find the LCD.
The LCD is
Step 2 Change each rational expression into anequivalent expression with the LCD. Then subtract.
Answer: C
Multiple-Choice Test Item
Find
A B
C D
Answer: C
Example 1 Mixed Expression to Rational Expression
Example 2 Complex Fraction Involving Numbers
Example 3 Complex Fraction Involving Monomials
Example 4 Complex Fraction Involving Polynomials
Simplify
The LCD is
Add the numerators.
Distributive Property
Answer: Simplify.
Simplify
Answer:
Baking Suppose Katelyn bought 2
pounds of chocolate chip cookie dough.
If the average cookie requires ounces of dough,
how many cookies would she be able to make?
To find the total number of cookies, divide the amount of cookie dough by the amount of dough needed for each cookie.
Convert pounds to ounces and divide by common units.
Simplify.
Express each term as an improper fraction.
Simplify.
Answer: Katelyn can make 21 cookies.
Answer: 27 cookies
Baking James bought pounds of cookie dough,
and he prefers to make large cookies. If each cookie
requires ounces of dough, how many cookies
can he make?
Simplify
Rewrite as a division sentence.
Rewrite as multiplication by the reciprocal.
Divide by common factors a, b, and c2.
a 4 c
2
b
3
1
1 1
Simplify.Answer:
Answer:
Simplify
Simplify
The LCD of the fractionsin the numerator is
Simplify the numerator.
The numerator contains a mixed expression. Rewrite it as a rational expression first.
Rewrite as a division sentence.
Multiply by the reciprocal of
Simplify.Answer:
Factor.
Simplify
Answer:
Example 1 Use Cross Products
Example 2 Use the LCD
Example 3 Multiple Solutions
Example 4 Work Problem
Example 5 Rate Problem
Example 6 No Solution
Example 7 Extraneous Solution
Solve
Original equation
Cross multiply.
Distributive Property
Add –2x and 48 to each side.
Answer: Divide each side by 6.
Answer: –3
Solve
Solve
Originalequation
The LCD is
DistributiveProperty
Simplify.
Add.
Subtract 1 from each side.
Divide each side by 6.
Answer:
Answer: 8
Solve
Solve
DistributiveProperty
Original equation
The LCD is
Simplify.
or
Set equal to 0.
Factor.
Check Check solutions by substituting each value in the original equation.
Check Check solutions by substituting each value in the original equation.
Answer: The number 1 is an excluded value for x. Thus, the solution is 3.
Solve
Answer: 4, –1
TV Installation On Saturdays, Lee helps her father
install satellite TV systems. The jobs normally take
Lee’s father about hours. But when Lee helps,
the jobs only take them hours. If Lee were
installing a satellite system herself, how long would
the job take?
Explore Since it takes Lee’s fatherhours
to install one job, he can finish of the job
in one hour. The amount of work Lee can
do in one hour can be represented by
To determine how long it takes Lee to do
the job, use the formula
Lee’s work + her father’s work = 1 job.
Plan The time that both of them worked was
hours. Each rate multiplied by this time results
in the amount of work done by each person.
Solve Lee’s her father’stotal
work plus workequals work.
1
Multiply.
The LCD is 10t.
Distributive Property
Simplify.
Add –6t to each side.
Divide each side by 4.
Answer: The job would take Lee or hours by herself.
Examine This seems reasonable because the combined efforts of the two took longer than half of her father’s usual time.
Driveways Shawna earns extra money by shoveling
driveways. If she works alone, she can finish a large
driveway in hours. If Vince helps her, they can get
done in hours. If Vince were shoveling the
driveway himself, how long would the job take him?
Answer: hours
Transportation The schedule for the Washington, D.C., Metrorail is shown to the right. Suppose two Red Line trains leave their stations at opposite ends of the line at exactly 2:00 P.M. One train travels between the two stations in 48 minutes and the other train takes 54 minutes. At what time do the two trains pass each other?
Determine the rates of both trains. The total distance is 19.4 miles.
Train 1 Train 2
Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 19.4 miles.
19.4
48
19.4
48
t19.4
54
19.4
54
t
t min Train 2
t min Train 1
d = r t
tr
The sum of the distances is 19.4.
The LCD is 432.
Distributive Property
Simplify.
Add.
Divide each sideby 329.8.
Answer: The trains passed each other at about 25 minutesafter they left their stations, at 2:25 P.M.
Transportation Two cyclists are riding on a 5-mile circular bike trail. They both leave the bike trail entrance at 3:00 P.M. traveling in opposite directions. It usually takes the first cyclist one hour to complete the trail and it takes the second cyclist 50 minutes. At what time will they pass each other?
Answer: 3:27 P.M.
Solve
Original equation
The LCD is x – 1.
Distributive Property1
1
1
1
Simplify.
Subtract 2 from each side.
Answer: Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus, the equation has no solution.
Solve
Answer: no solutions
Solve
Original equation
The LCD is x – 2.
1
1
1
1
Distributive Property
Simplify.
Subtract 4 fromeach side.
Factor.
or Zero Product Property
Answer: The number 2 is an extraneous solution, since 2 is an excluded value for x. Thus, –2 is the solution of the equation.
Solve
Answer: –3
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