Download - TEXT 12. Rational Expressions_eq
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ALGEBRA PROJECT
UNIT 12
RATIONAL EXPRESSIONS and EQUATIONS
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RATIONAL EXPRESSIONS and EQUATIONS
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Lesson 1 Inverse Variation
Lesson 2 Rational Expressions
Lesson 3 Multiplying Rational Expressions
Lesson 4 Dividing Rational Expressions
Lesson 5 Dividing Polynomials
Lesson 6Rational Expressions with Like Denominators
Lesson 7Rational Expressions with Unlike Denominators
Lesson 8 Mixed Expressions and Complex Fractions
Lesson 9 Solving Rational Equations
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INVERSE VARIATION
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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
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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.
t
12108642p
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t
12108642p
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
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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.
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t
12108642p
33.64.56918t
12108642pAnswer:
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.
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Answer:
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Inverse variation equation
The constant of variation is 4.
Graph an inverse variation in which y varies inversely as x and
Solve for k.
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14
22
41
undefined0
–4–1
–2–2
–1–4
yx
Choose values for x and y whose product is 4.
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Answer:
Graph an inverse variation in which y varies inversely as x and
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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
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Method 2 Use a proportion.
Proportion rule for inverse variations
Cross multiply.
Divide each side by 15.
Answer: Both methods show that
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If y varies inversely as x andfind x when
Answer: 8
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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:
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If y varies inversely as x and find y when
Answer: –25
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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?
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Original equation
Divide each side by 2.
Simplify.
Answer: The 2-kilogram weight should be 9.6 meters from the fulcrum.
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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?
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RATIONAL EXPRESSIONS
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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
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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
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Answer: –3
State the excluded value of
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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 value of
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Answer: 2, 3
State the excluded value of
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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.
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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.
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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?
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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
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The GCF of the numeratorand denominator is
Divide the numerator anddenominator by
1
1
Simplify
Answer: Simplify.
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Simplify
Answer:
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Factor.
Divide the numerator and denominator by the GCF, x – 7.
1
1
Simplify
Answer: Simplify
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Simplify
Answer:
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Divide the numeratorand denominator bythe
1
1
Simplify State the excluded values of x.
Factor.
Simplify.Answer:
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Exclude the values for which equals 0.
The denominator cannot equal zero.
Factor.
Zero Product Property
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Evaluate.
Simplify.
Check Verify the excluded values by substituting them into the original expression.
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Evaluate.
Simplify.
Answer: The expression is undefined when andTherefore,
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Answer:
Simplify State the excluded values of w.
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MULTIPLYINGRATIONAL EXPRESSIONS
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Example 1 Expressions Involving Monomials
Example 2 Expressions Involving Polynomials
Example 3 Dimensional Analysis
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Method 1 Divide by the greatest common factor after multiplying.
Multiply the numerators.
Multiply the denominators.
The GCF is 98xyz.
Simplify.
Find
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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
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Multiply.Answer:
Divide bycommon factors and r.
1 1 1
1 1 2 1 1
d 2 q
2 r 3
Find
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Answer:
Answer:
a. Find
b. Find
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Factor thenumerator.
Simplify.Answer:
The GCF is1
1
1
x 2
Find
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Find
Factor.
The GCF is1
1
1
1
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Multiply.
Simplify.Answer:
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a. Find
b. Find
Answer:
Answer:
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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?
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Answer: The escape velocity is 11,200 meters per second.
1120 10
1 1
Simplify.
Multiply.
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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
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DIVIDINGRATIONAL EXPRESSIONS
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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
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Find
Multiply by
the reciprocal of
Answer: Simplify.
Divide by common factors 5, 6, and x.
1
4 11
15x3
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Find
Answer:
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Find
Multiply by
the reciprocal
of
Factor
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or Simplify.Answer:
The GCFis
1
1
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Find
Answer:
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Find
Multiply by
the reciprocal
of
Factor
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Simplify.Answer:
The GCF is
1
1
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Find
Answer:
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Find
Multiply by the
reciprocal,
Factor
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Simplify.Answer:
The GCFis
1
1
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Find
Answer:
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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.
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Convert days to hours.
Answer: Thus, the speed of the aircraft was about 116 miles per hour.
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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
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DIVIDING POLYNOMIALS
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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
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Find
Write as a rational expression.
Divide each term by 2x.
2x 9
11
Simplify each term.
Simplify.Answer:
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Find
Answer:
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Find
Write as a rational expression.
Divide each term by 3y.
Simplify.Answer:
2y 1
13
Simplify each term.
y
3
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Find
Answer:
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1
1
Divide by the GCF.
Find
Factor the numerator.
Simplify.Answer:
Write as a rational expression.
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Find
Answer:
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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.
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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.
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Answer: The quotient of is
with a remainder of 3, which can be written as
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Find
Answer: The quotient is with a remainder of 2.
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Rename the x2 term by using a coefficient of 0.
Find
![Page 83: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/83.jpg)
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:
![Page 84: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/84.jpg)
Find
Answer: The quotient is
![Page 85: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/85.jpg)
RATIONAL EXPRESSIONS WITH LIKE DENOMINATORS
![Page 86: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/86.jpg)
Example 1 Numbers in Denominator
Example 2 Binomials in Denominator
Example 3 Find a Perimeter
Example 4 Subtract Rational Expressions
Example 5 Inverse Denominators
![Page 87: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/87.jpg)
Find
The common denominator is 15.
Add the numerators.
Simplify.Answer:
Divide by the common factor, 5.
4b
3
![Page 88: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/88.jpg)
Find
Answer:
![Page 89: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/89.jpg)
Divide by the common factor, c + 2.
1
1
Find
The common denominator is c + 2.
Factor the numerator.
Simplify.Answer:
![Page 90: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/90.jpg)
Answer: 5
Find
![Page 91: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/91.jpg)
Geometry Find an expression for the perimeter of rectangle WXYZ.
Perimeter formula
![Page 92: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/92.jpg)
The commondenominator is
Distributive Property
Combine like terms.
Factor.
![Page 93: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/93.jpg)
Answer: The perimeter can be represented by the
expression
![Page 94: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/94.jpg)
Geometry Find an expression for the perimeter of rectangle PQRS.
Answer:
![Page 95: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/95.jpg)
Find
The common denominator is
The additive inverse ofis
Distributive Property
![Page 96: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/96.jpg)
Simplify.Answer:
![Page 97: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/97.jpg)
Find
Answer:
![Page 98: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/98.jpg)
Find
The denominator is the same asor . Rewrite the second expression so that it has the same denominator as the first.
Rewrite usingcommon denominators.
![Page 99: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/99.jpg)
The common denominator is
Simplify.Answer:
![Page 100: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/100.jpg)
Find
Answer:
![Page 101: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/101.jpg)
RATIONAL EXPRESSIONS WITH UNLIKE DENOMINATORS
![Page 102: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/102.jpg)
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
![Page 103: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/103.jpg)
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:
![Page 104: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/104.jpg)
Find the LCM of
Answer:
![Page 105: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/105.jpg)
Find the LCM of
Express each polynomial in factored form.
Use each factor the greatest number of times it appears.
Answer:
![Page 106: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/106.jpg)
Find the LCM of
Answer:
![Page 107: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/107.jpg)
Find
Factor each denominator and find the LCD.
Since the denominator of is already 5z, onlyneeds to be renamed.
![Page 108: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/108.jpg)
Multiply
by
Distributive Property
Add the numerators.
Answer: Simplify.
![Page 109: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/109.jpg)
Find
Answer:
![Page 110: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/110.jpg)
Find
Factor thedenominators.
The LCDis
![Page 111: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/111.jpg)
Add thenumerators.
Simplify.Answer:
![Page 112: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/112.jpg)
Find
Answer:
![Page 113: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/113.jpg)
Find
Factor.
The LCD is
Add thenumerators.
![Page 114: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/114.jpg)
Multiply.
Simplify.Answer:
![Page 115: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/115.jpg)
Find
Answer:
![Page 116: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/116.jpg)
Multiple-Choice Test Item
Find
A B
C D
![Page 117: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/117.jpg)
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
![Page 118: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/118.jpg)
Step 2 Change each rational expression into anequivalent expression with the LCD. Then subtract.
![Page 119: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/119.jpg)
Answer: C
![Page 120: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/120.jpg)
Multiple-Choice Test Item
Find
A B
C D
Answer: C
![Page 121: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/121.jpg)
MIXED EXPRESSIONS andCOMPLEX FRACTIONS
![Page 122: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/122.jpg)
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
![Page 123: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/123.jpg)
Simplify
The LCD is
Add the numerators.
Distributive Property
Answer: Simplify.
![Page 124: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/124.jpg)
Simplify
Answer:
![Page 125: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/125.jpg)
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.
![Page 126: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/126.jpg)
Convert pounds to ounces and divide by common units.
Simplify.
Express each term as an improper fraction.
![Page 127: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/127.jpg)
Simplify.
Answer: Katelyn can make 21 cookies.
![Page 128: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/128.jpg)
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?
![Page 129: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/129.jpg)
Simplify
Rewrite as a division sentence.
Rewrite as multiplication by the reciprocal.
![Page 130: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/130.jpg)
Divide by common factors a, b, and c2.
a 4 c
2
b
3
1
1 1
Simplify.Answer:
![Page 131: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/131.jpg)
Answer:
Simplify
![Page 132: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/132.jpg)
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.
![Page 133: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/133.jpg)
Rewrite as a division sentence.
Multiply by the reciprocal of
Simplify.Answer:
Factor.
![Page 134: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/134.jpg)
Simplify
Answer:
![Page 135: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/135.jpg)
SOLVINGRATIONAL EQUATIONS
![Page 136: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/136.jpg)
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
![Page 137: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/137.jpg)
Solve
Original equation
Cross multiply.
Distributive Property
Add –2x and 48 to each side.
Answer: Divide each side by 6.
![Page 138: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/138.jpg)
Answer: –3
Solve
![Page 139: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/139.jpg)
Solve
Originalequation
The LCD is
DistributiveProperty
![Page 140: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/140.jpg)
Simplify.
Add.
Subtract 1 from each side.
Divide each side by 6.
Answer:
![Page 141: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/141.jpg)
Answer: 8
Solve
![Page 142: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/142.jpg)
Solve
DistributiveProperty
Original equation
The LCD is
![Page 143: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/143.jpg)
Simplify.
or
Set equal to 0.
Factor.
![Page 144: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/144.jpg)
Check Check solutions by substituting each value in the original equation.
![Page 145: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/145.jpg)
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.
![Page 146: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/146.jpg)
Solve
Answer: 4, –1
![Page 147: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/147.jpg)
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?
![Page 148: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/148.jpg)
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.
![Page 149: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/149.jpg)
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 work equals work.
1
![Page 150: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/150.jpg)
Multiply.
The LCD is 10t.
Distributive Property
Simplify.
![Page 151: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/151.jpg)
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.
![Page 152: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/152.jpg)
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
![Page 153: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/153.jpg)
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?
![Page 154: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/154.jpg)
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.
![Page 155: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/155.jpg)
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.
![Page 156: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/156.jpg)
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.
![Page 157: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/157.jpg)
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.
![Page 158: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/158.jpg)
Solve
Original equation
The LCD is x – 1.
Distributive Property1
1
1
1
![Page 159: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/159.jpg)
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.
![Page 160: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/160.jpg)
Solve
Answer: no solutions
![Page 161: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/161.jpg)
Solve
Original equation
The LCD is x – 2.
1
1
1
1
Distributive Property
![Page 162: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/162.jpg)
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.
![Page 163: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/163.jpg)
Solve
Answer: –3
![Page 164: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/164.jpg)
THIS IS THE ENDOF THE SESSION
![Page 165: TEXT 12. Rational Expressions_eq](https://reader033.vdocuments.mx/reader033/viewer/2022061117/54665e74b4af9f8b358b4edc/html5/thumbnails/165.jpg)
BYE!