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Section 7.4 Adding and Subtracting Rational Expressions 383
Adding and Subtracting Rational Expressions
Work with a partner. Find the sum or difference of the two rational expressions. Then match the sum or difference with its domain. Explain your reasoning. Sum or Difference Domain
a.
— x − 1
— x − 1
= A. all real numbers except − 2
b.
— x − 1
— x
= B. all real numbers except −1 and 1
c.
— x − 2
— 2 − x
= C. all real numbers except 1
d.
— x − 1
— x + 1
= D. all real numbers except 0
e. x — x + 2
x + 1 — 2 + x
= E. all real numbers except −2 and 1
f. x — x − 2
x + 1 — x
= F. all real numbers except 0 and 1
g. x — x + 2
x — x − 1
= G. all real numbers except 2
h. x + 2 — x
x + 1 — x
= H. all real numbers except 0 and 2
Writing a Sum or Difference
Work with a partner. Write a sum or difference of rational expressions that has the given domain. Justify your answer. a. all real numbers except − 1 b. all real numbers except −1 and 3 c. all real numbers except −1, 0, and 3
Communicate Your AnswerCommunicate Your Answer
3. How can you determine the domain of the sum or difference of two rational expressions? 4. Your friend found a sum as follows. Describe and correct the error(s). —^ x x + 4
— x − 4
x + 3 — 2 x
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in math, you need to justify your conclusions and communicate them to others.
Essential QuestionEssential Question How can you determine the domain of the sum
or difference of two rational expressions?
You can add and subtract rational expressions in much the same way that you add and subtract fractions. —^ x x + 1
+ —^2
x + 1
= — x^ +^2 x + 1
Sum of rational expressions
x
− —^1
2 x
= —^2
2 x
− —^1
2 x
= —^1
2 x
Difference of rational expressions
Adding and Subtracting
Rational Expressions
384 Chapter 7 Rational Functions
7.4 Lesson^ What You Will LearnWhat You Will Learn
Add or subtract rational expressions. Rewrite rational expressions and graph the related function. Simplify complex fractions.
Adding or Subtracting Rational Expressions As with numerical fractions, the procedure used to add (or subtract) two rational expressions depends upon whether the expressions have like or unlike denominators. To add (or subtract) rational expressions with like denominators, simply add (or subtract) their numerators. Then place the result over the common denominator.
Adding or Subtracting with Like Denominators
a.
— 4 x
— 4 x
— 4 x
— 4 x
— 2 x Add numerators and simplify.
b. 2 x — x + 6
— x + 6
2 x − 5 — x + 6 Subtract numerators.
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Find the sum or difference.
1.
— 12 x
— 12 x
— 3 x^2
— 3 x^2
4 x — x − 2
x — x − 2
2 x^2 — x^2 + 1
— x^2 + 1
To add (or subtract) two rational expressions with unlike denominators, find a common denominator. Rewrite each rational expression using the common denominator. Then add (or subtract).
complex fraction, p. 387 Previous rational numbers reciprocal
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
CoreCore ConceptConcept
Adding or Subtracting with Like Denominators
Let a , b , and c be expressions with c ≠ 0. Addition Subtraction a — c
b — c
a + b — c
a — c
b — c
a − b — c
Adding or Subtracting with Unlike Denominators
Let a , b , c , and d be expressions with c ≠ 0 and d ≠ 0. Addition Subtraction
a — c
= — ad cd
= ad —^ +^ bc cd
a — c
− b — d
= — ad cd
− — bc cd
= ad —^ −^ bc cd
You can always find a common denominator of two rational expressions by multiplying the denominators, as shown above. However, when you use the least common denominator (LCD), which is the least common multiple (LCM) of the denominators, simplifying your answer may take fewer steps.
386 Chapter 7 Rational Functions
Subtracting with Unlike Denominators
Find the difference — x^ +^2 2 x − 2
− —−^2 x^ −^1 x^2 − 4 x + 3
SOLUTION
—^ x^ +^2 2 x − 2
− —−^2 x^ −^1 x^2 − 4 x + 3
= — x^ +^2 2( x − 1)
− ——−^2 x^ −^1 ( x − 1)( x − 3)
Factor each denominator.
= — x^ +^2 2( x − 1) ⋅
x — − 3 x − 3
− ——−^2 x^ −^1 ( x − 1)( x − 3) ⋅
LCD is 2( x − 1)( x − 3).
= x
(^2) − x − 6 —— 2( x − 1)( x − 3)
− ——−^4 x^ −^2 2( x − 1)( x − 3)
Multiply.
= x
(^2) − x − 6 − (− 4 x − 2) ——— 2( x − 1)( x − 3)
Subtract numerators.
= x
(^2) + 3 x − 4 —— 2( x − 1)( x − 3)
Simplify numerator.
= ——( x^ −^ 1)( x^ +^ 4) 2( x − 1)( x − 3)
Factor numerator. Divide out common factors. = — x^ +^4 2( x − 3)
, x ≠ − 1 Simplify.
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5. Find the least common multiple of 5 x^3 and 10 x^2 − 15 x. Find the sum or difference. 6. (^) —^3 4 x
− —^1
7. —^1
3 x^2
8. (^) — x x^2 − x − 12
+ —^5
12 x − 48
Rewriting Rational Functions Rewriting a rational expression may reveal properties of the related function and its graph. In Example 4 of Section 7.2, you used long division to rewrite a rational expression. In the next example, you will use inspection.
Rewriting and Graphing a Rational Function
Rewrite the function g ( x ) = 3 — x^ +^5 x + 1
in the form g ( x ) = — a x − h
- k. Graph the function. Describe the graph of g as a transformation of the graph of f ( x ) = — a x
SOLUTION
Rewrite by inspection: 3 x + 5 — x + 1
= (^3) — x^ +^3 +^2 x + 1
= 3( —— x^ +^ 1)^ +^2 x + 1
= 3( — x^ +^ 1) x + 1
+ —^2
x + 1
= 3 + —^2
x + 1
The rewritten function is g ( x ) = —^2 x + 1
- The graph of g is a translation 1 unit
left and 3 units up of the graph of f ( x ) = 2 — x
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9. Rewrite g ( x ) = —^2 x^ −^4 x − 3
in the form g ( x ) = — a x − h
- k. Graph the function. Describe the graph of g as a transformation of the graph of f ( x ) = — a x
COMMON ERROR
When subtracting rational expressions, remember to distribute the negative sign to all the terms in the quantity that is being subtracted.
x
y
− 4 2
2
4
g
Section 7.4 Adding and Subtracting Rational Expressions 387
Complex Fractions
A complex fraction is a fraction that contains a fraction in its numerator or denominator. A complex fraction can be simplified using either of the methods below.
Simplifying a Complex Fraction
Simplify
— x + 4 — —^1 x + 4
x
SOLUTION
Method 1
— x + 4 — —^1 x + 4
x
— x + 4 — —^3 x^ +^8 x ( x + 4)
Add fractions in denominator.
= —^5
x + 4 ⋅
x —( x + 4) 3 x + 8
Multiply by reciprocal.
= ——^5 x ( x^ +^ 4) ( x + 4)(3 x + 8)
Divide out common factors.
= —^5 x 3 x + 8
, x ≠ −4, x ≠ 0 Simplify.
Method 2 The LCD of all the fractions in the numerator and denominator is x ( x + 4).
5 — x + 4 — —^1 x + 4
x
— x + 4 — —^1 x + 4
+ —^2
x
⋅
x —( x + 4) x ( x + 4)
Multiply numerator and denominator by the LCD.
— x + 4 ⋅^
x ( x + 4) ——— —^1 x + 4 ⋅^
x ( x + 4) + 2 — x ⋅^
x ( x + 4)
Divide out common factors.
= ——^5 x x + 2( x + 4)
Simplify.
= —^5 x 3 x + 8
, x ≠ −4, x ≠ 0 Simplify.
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Simplify the complex fraction.
x — 6
− x — 3 — —^ x 5
− —^7
— x
— 2 — x
— x + 5 —— 2 — x − 3
+ —^1
x + 5
CoreCore ConceptConcept
Simplifying Complex Fractions
Method 1 If necessary, simplify the numerator and denominator by writing each as a single fraction. Then divide by multiplying the numerator by the reciprocal of the denominator. Method 2 Multiply the numerator and the denominator by the LCD of every fraction in the numerator and denominator. Then simplify.
Section 7.4 Adding and Subtracting Rational Expressions 389
In Exercises 31–38, rewrite the function g in the form
g ( x ) = a — x − h +^ k.^ Graph the function. Describe the
graph of g as a transformation of the graph of f ( x ) = a — x****. (See Example 5.)
31. g ( x ) = 5 x − 7 — x − 1 32. g ( x ) = 6 x + 4 — x + 5 33. g ( x ) = 12 x — x − 5 34. g ( x ) = 8 x — x + 13 35. g ( x ) = 2 x + 3 — x 36. g ( x ) = 4 x − 6 — x 37. g ( x ) = 3 x + 11 — x − 3 38. g ( x ) = 7 x − 9 — x + 10
In Exercises 39–44, simplify the complex fraction. (See Example 6.)
x — 3
— 10 +
— x
x — x — 5
—^1
2 x − 5
— 8 x − 20 —— x — 2 x − 5
—^16
x − 2 — —^4 x + 1
x
—^1
3 x^2 − 3 —— —^5 x + 1
− — x^ +^4 x^2 − 3 x − 4
—^3
x − 2
− —^6
x^2 − 4 —— 3 — x + 2
— x − 2
45. PROBLEM SOLVING The total time T (in hours) needed to fl y from New York to Los Angeles and back can be modeled by the equation below, where d is the distance (in miles) each way, a is the average airplane speed (in miles per hour), and j is the average speed (in miles per hour) of the jet stream. Simplify the equation. Then find the total time it takes to fly 2468 miles when a = 510 miles per hour and j = 115 miles per hour.
T = d — a − j
d — a + j
LA
NY
LA a a − j
j
LA
NY
LA A a a + j
j
46. REWRITING A FORMULA The total resistance Rt of two resistors in a parallel circuit with resistances R 1 and R 2 (in ohms) is given by the equation shown. Simplify the complex fraction. Then find the total resistance when R 1 = 2000 ohms and R 2 = 5600 ohms.
Rt =
— —^1 R 1
+ —^1
R 2
R (^) t
R 1
R 2
47. PROBLEM SOLVING You plan a trip that involves a 40-mile bus ride and a train ride. The entire trip is 140 miles. The time (in hours) the bus travels is y 1 = (^) —^40 x
, where x is the average speed (in miles per hour) of the bus. The time (in hours) the train travels is y 2 = (^) —^100 x + 30
. Write and simplify a model that shows the total time y of the trip. 48. PROBLEM SOLVING You participate in a sprint triathlon that involves swimming, bicycling, and running. The table shows the distances (in miles) and your average speed for each portion of the race.
Distance (miles)
Speed (miles per hour)
Swimming 0.5 r
Bicycling 22 15 r
Running 6 r + 5
a. Write a model in simplified form for the total time (in hours) it takes to complete the race. b. How long does it take to complete the race if you can swim at an average speed of 2 miles per hour? Justify your answer.
49. MAKING AN ARGUMENT Your friend claims that the least common multiple of two numbers is always greater than each of the numbers. Is your friend correct? Justify your answer.
390 Chapter 7 Rational Functions
50. HOW DO YOU SEE IT?
Use the graph of the function f ( x ) = — a x − h
- k to determine the values of h and k.
51. REWRITING A FORMULA You borrow P dollars to buy a car and agree to repay the loan over t years at a monthly interest rate of i (expressed as a decimal). Your monthly payment M is given by either formula below.
M = Pi —— 1 − (^) (
— 1 + i )
12 t or M = Pi (1^ +^ i )
12 t —— (1 + i )^12 t^ − 1
a. Show that the formulas are equivalent by simplifying the fi rst formula. b. Find your monthly payment when you borrow $15,500 at a monthly interest rate of 0.5% and repay the loan over 4 years.
52. THOUGHT PROVOKING Is it possible to write two rational functions whose sum is a quadratic function? Justify your answer. 53. USING TOOLS Use technology to rewrite the function g ( x ) = ———(97.6)(0.024)^ +^ x (0.003) 12.2 + x
in the form g ( x ) = — a x − h
- k. Describe the graph of g as a transformation of the graph of f ( x ) = a — x
54. MATHEMATICAL CONNECTIONS Find an expression for the surface area of the box.
x + 5 x
x x + 1
x + 1 3
55. PROBLEM SOLVING You are hired to wash the new cars at a car dealership with two other employees. You take an average of 40 minutes to wash a car ( R 1 = 1/40 car per minute). The second employee washes a car in x minutes. The third employee washes a car in x + 10 minutes. a. Write expressions for the rates that each employee can wash a car. b. Write a single expression R for the combined rate of cars washed per minute by the group. c. Evaluate your expression in part (b) when the second employee washes a car in 35 minutes. How many cars per hour does this represent? Explain your reasoning. 56. MODELING WITH MATHEMATICS The amount A (in milligrams) of aspirin in a person’s bloodstream can be modeled by
A = 391 t
—— 0.218 t^4 + 0.991 t^2 + 1 where t is the time (in hours) after one dose is taken.
A A first dose second dose
combined effect
a. A second dose is taken 1 hour after the first dose. Write an equation to model the amount of the second dose in the bloodstream. b. Write a model for the total amount of aspirin in the bloodstream after the second dose is taken.
57. FINDING A PATTERN Find the next two expressions in the pattern shown. Then simplify all five expressions. What value do the expressions approach?
1 +
— 2 + 1 — 2
— 2 +
— 2 +
— 2
—— 2 +
— 2 +
— 2 +
— 2
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Solve the system by graphing. (Section 3.5)
58. y = x^2 + 6 59. 2 x^2 − 3 x − y = 0 60. 3 = y − x^2 − x 61. y = ( x + 2)^2 − 3 y = 3 x + 4 —^52 x − y = 9 — 4 y = − x^2 − 3 x − 5 y = x^2 + 4 x + 5
Reviewing what you learned in previous grades and lessons
x
y
− 4 − 2 2
− 4
− 6
f
