Chapt 6. Rational Expressions, Functions, and Equations

Chapt 6. Rational Chapt 6. Rational Expressions, Functions, Expressions, Functions,

and Equationsand Equations

6.1 Rational Expressions and 6.1 Rational Expressions and FunctionsFunctions

Rational ExpressionRational Expression Polynomial divided by non-zero polynomialPolynomial divided by non-zero polynomial 120x / (100 – x)120x / (100 – x)

(3x(3x22 - 12xy – 15y - 12xy – 15y22) / (6x) / (6x33 – 6xy – 6xy22)) Rational FunctionRational Function

Function defined by a rational expressionFunction defined by a rational expression f(x) = (120x) / (100 – x)f(x) = (120x) / (100 – x)

Evaluating a FunctionEvaluating a Function

Given: f(x) = 120x / (100 – x)Given: f(x) = 120x / (100 – x) Evaluate: f(20)Evaluate: f(20)

f(20) = 120(20) / (100 – (20))f(20) = 120(20) / (100 – (20)) = 2400 / 80 = 2400 / 80 = 30 = 30

f(40) = 120(40) / (100 – (40))f(40) = 120(40) / (100 – (40)) = 4800 / 60 = 4800 / 60 = 80 = 80

Domain of a Rational FunctionDomain of a Rational Function Given: Given:

The cost (in $1000) of cleaning up a The cost (in $1000) of cleaning up a polluted lake is a function of the percentage polluted lake is a function of the percentage (x) of the lake’s pollutants to be removed. It (x) of the lake’s pollutants to be removed. It is given by the following function.is given by the following function.

f(x) = 120x / (100 – x)f(x) = 120x / (100 – x)

What is the cost of cleaning up 50% of the What is the cost of cleaning up 50% of the pollutants?pollutants? f(50) = 120(50) / (100 – 50) = 120f(50) = 120(50) / (100 – 50) = 120

Domain of a Rational FunctionDomain of a Rational Function

Given the last function: Given the last function:

f(x) = 120x / (100 – x)f(x) = 120x / (100 – x)

What are the possible values of x?What are the possible values of x? Answer:Answer:

x x ≠≠ 100 100 x cannot be negative (in practical cases)x cannot be negative (in practical cases)

Domain ofDomain of f f:: [0, 2) U (2, 100][0, 2) U (2, 100]

Domain of a Rational FunctionDomain of a Rational Function

Given: f(x) = (2x + 1) / (2xGiven: f(x) = (2x + 1) / (2x22 – x – 1) – x – 1) What is the domain of f?What is the domain of f? Solution:Solution:

(2x(2x2 – x – 1) – x – 1)(2x + 1)(x – 1) = 0(2x + 1)(x – 1) = 02x + 1 = 0 x – 1 = 02x + 1 = 0 x – 1 = 0x = -1/2 x = 1x = -1/2 x = 1

Domain of f: Domain of f: (-(-∞∞ , -1/2) U (-1/2, 1) U (1, , -1/2) U (-1/2, 1) U (1, ∞∞))

-1/2 1

Your TurnYour Turn

Given: f(x) = (x – 5) / (2xGiven: f(x) = (x – 5) / (2x22 + 5x – 3) + 5x – 3) Find the domain of f.Find the domain of f. Solution:Solution:

2x 2x22 + 5x – 3 + 5x – 3(2x - 1)(x + 3) = 0(2x - 1)(x + 3) = 02x – 1 = 0 x + 3 = 02x – 1 = 0 x + 3 = 0x = ½ x = -3x = ½ x = -3

Domain of f:Domain of f:(-(-∞∞ , -3) U (-3, 1/2) U (1/2, , -3) U (-3, 1/2) U (1/2, ∞∞))

Simplifying Rational ExpressionsSimplifying Rational Expressions

Simplify: (xSimplify: (x22 + 4x + 3) / (x + 1) + 4x + 3) / (x + 1)

x x22 + 4x + 3 (x + 1)(x + 3) + 4x + 3 (x + 1)(x + 3)--------------- = ------------------ = x + 1, x --------------- = ------------------ = x + 1, x ≠≠ -1 -1 x + 1 (x + 1) x + 1 (x + 1)

y = x + 1 y = (x2 + 4x + 3)/(x + 1)

Your TurnYour Turn

SimplifySimplify1.1. (x(x2 2 + 7x + 10) / (x + 2)+ 7x + 10) / (x + 2) = (x + 2)(x + 5) / (x + 2)= (x + 2)(x + 5) / (x + 2)

= x + 5, x = x + 5, x ≠≠ -2 -2

2.2. (x(x22 – 7x – 18) / (2x – 7x – 18) / (2x22 + 3x – 2) + 3x – 2) = (x + 2)(x – 9) / (2x - 1)(x + 2)= (x + 2)(x – 9) / (2x - 1)(x + 2)

= (x – 9) / (2x – 1), x = (x – 9) / (2x – 1), x ≠≠ -2 and x -2 and x ≠≠ 1/2 1/2

Multiplying Rational ExpressionsMultiplying Rational Expressions MultiplyMultiply

x + 4 xx + 4 x2 – 4x - 21 – 4x - 21 -------- -------- ∙ ----------------∙ ---------------- x – 7 x x – 7 x2 – 16 – 16

x + 4 (x – 7)(x + 3)x + 4 (x – 7)(x + 3)= -------- = -------- · -------------------· ------------------- x – 7 (x – 4)(x + 4) x – 7 (x – 4)(x + 4)

x + 3x + 3= --------= --------

x – 4 x – 4

Dividing Rational ExpressionsDividing Rational Expressions DivideDivide

(y(y2 – 25) / (2y – 2) (y – 25) / (2y – 2) (y2 + 10y +25) / (y + 10y +25) / (y2 + 4y – 5)+ 4y – 5) = (y= (y2 – 25) / (2y – 2) – 25) / (2y – 2) ∙∙ (y (y2 + 4y – 5)/(y+ 4y – 5)/(y2 + 10y + 25) + 10y + 25)

(y – 5)(y + 5) (y + 5)(y – 1) (y – 5)(y + 5) (y + 5)(y – 1) = ------------------ ∙ -------------------= ------------------ ∙ -------------------

2(y – 1) (y + 5)(y + 5) 2(y – 1) (y + 5)(y + 5)

y - 5y - 5 = --------= --------

2 2

Your TurnYour Turn Simplify the followingSimplify the following

xx2 + xy 4x – 4y + xy 4x – 4y ----------- ----------- ·· ---------- ----------xx2 – y – y2 x x

x(x + y) 4(x – y) x(x + y) 4(x – y) = ------------------ = ------------------ · ------------· ------------ (x – y)(x + y) x (x – y)(x + y) x

= 4= 4

Your TurnYour Turn

SimplifySimplify (y(y2 – 4) / (y – 4) / (y2 + y) (y + y) (y2 + 5y + 6) / (y + 5y + 6) / (y2 – 1) – 1)

= (y= (y2 – 4) / (y – 4) / (y2 + y) + y) ∙∙ (y (y2 – 1) / (y – 1) / (y2 + 5y + 6) + 5y + 6)

(y – 2)(y + 2) (y - 1)(y + 1)(y – 2)(y + 2) (y - 1)(y + 1)= -------------------- = -------------------- ∙∙ ------------------ ------------------ y(y + 1) (y + 2)(y + 3) y(y + 1) (y + 2)(y + 3)

(y – 2)(y – 1)(y – 2)(y – 1)= -------------------= ------------------- y(y + 3) y(y + 3)

6.2 Adding and Subtracting 6.2 Adding and Subtracting Rational ExpressionsRational Expressions

AddAdd xx2 + 2x – 2 5x + 12 + 2x – 2 5x + 12

------------------- + ------------------ ------------------- + ------------------ x x2 + 3x – 10 x + 3x – 10 x2 + 3x – 10 + 3x – 10

xx2 + 2x – 2 + 5x + 12 x + 2x – 2 + 5x + 12 x2 + 7x + 10 + 7x + 10 = ---------------------------- = --------------------= ---------------------------- = --------------------

x x2 + 3x – 10 x + 3x – 10 x2 + 3x – 10 + 3x – 10 (x + 2) (x + 5) (x + 2) (x + 2) (x + 5) (x + 2)

= -------------------- = -------------= -------------------- = ------------- (x + 5)(x – 2) (x – 2) (x + 5)(x – 2) (x – 2)

Your TurnYour Turn AddAdd

xx2 + 5x – 15 -2x + 5 + 5x – 15 -2x + 5 ------------------- + ------------------ ------------------- + ------------------ x x2 + 5x + 6 x + 5x + 6 x2 + 5x + 6 + 5x + 6

xx2 + 5x – 15 - 2x + 5 x + 5x – 15 - 2x + 5 x2 + 3x – 10 + 3x – 10 = ------------------------------ = --------------------= ------------------------------ = --------------------

x x2 + 5x + 6 x + 5x + 6 x2 + 5x + 6 + 5x + 6

(x - 2) (x + 5) (x - 2) (x + 5) = -------------------- = --------------------

(x + 2)(x + 3) (x + 2)(x + 3)

Your TurnYour Turn SubtractSubtract

3y3y3 – 5x – 5x3 4y 4y3 – 6x – 6x3

--------------- - --------------- --------------- - --------------- x x2 – y – y2 x x2 – y – y2

3y3y3 – 5x – 5x3 - (4y - (4y3 – 6x – 6x3) 3y3y3 – 5x – 5x3 - 4y - 4y3 + 6x + 6x3

= ------------------------------- = ----------------------------= ------------------------------- = ---------------------------- x x2 – y – y2 xx2 – y – y2

xx3 - y - y3 (x – y)(x2 + xy + y2) (x2 + xy + y2)

= ---------------- = --------------------------- = --------------------= ---------------- = --------------------------- = -------------------- x x2 – y – y2 (x – y)(x + y) (x + y)(x – y)(x + y) (x + y)

Finding the Least Common Finding the Least Common DenominatorDenominator

Find the LCD of: 7/6xFind the LCD of: 7/6x2 & 2/9x & 2/9x Solution:Solution:

1.1. Factor denominatorsFactor denominators6x6x2 2, 3, x, x 2, 3, x, x9x 9x 3, 3, x 3, 3, x

2.2. List all factors of 1List all factors of 1stst Denominator—2, 3, x, x Denominator—2, 3, x, x

3.3. Add factors of 2Add factors of 2ndnd dominator not in the list dominator not in the list—2, 3, x, x, & 3—2, 3, x, x, & 3

4.4. LCD: product of all factors in the list—18xLCD: product of all factors in the list—18x2

Finding the Least Common Finding the Least Common DenominatorDenominator

Find the LCD of: Find the LCD of: 7/(5x7/(5x2 + 15x) and 9/(x + 15x) and 9/(x2 + 6x + 9) + 6x + 9)

Solution:Solution:1.1. Find factors in 1Find factors in 1stst denominator denominator

5x5x2 + 15x + 15x 5x(x + 3) 5x(x + 3)

2.2. Find factors of 2Find factors of 2ndnd denominator denominatorxx2 + 6x + 9 + 6x + 9 (x + 3)(x + 3) (x + 3)(x + 3)

3.3. List factors of 1List factors of 1stst denominator denominator5x(x + 3)5x(x + 3)

4.4. Include in the list those factors in 2Include in the list those factors in 2ndnd denominator not denominator not found in 1stfound in 1st5x(x + 3)(x + 3) or 5x(x + 3)(x + 3) or 5x(x + 3)5x(x + 3)2

Your TurnYour Turn

Find the LCD of:Find the LCD of:1.1. 7 / (y7 / (y2 – 4) and 15 / (y – 4) and 15 / (y2 + 2y) + 2y)

• 11stst den: y den: y22 – 4 = (y + 2)(y – 2) – 4 = (y + 2)(y – 2)• 22ndnd den: y den: y22 + 2y = y(y + 2) + 2y = y(y + 2)• LCD: (y + 2)(y – 2)yLCD: (y + 2)(y – 2)y

2.2. 3/(y3/(y2 – 5y – 6) and 6/(y – 5y – 6) and 6/(y2 – 4y – 5) – 4y – 5)• 11stst den: y den: y22 – 5y – 6 = (y – 6)(y + 1) – 5y – 6 = (y – 6)(y + 1)• 22ndnd den: y den: y22 – 4y – 5 = (y – 5)(y + 1) – 4y – 5 = (y – 5)(y + 1)• LCD: (y – 6)(y + 1)(y – 5)LCD: (y – 6)(y + 1)(y – 5)

6.3 Complex Rational Expressions6.3 Complex Rational Expressions

Given:Given: p =principal (amount borrowed)p =principal (amount borrowed) r = monthly interest rater = monthly interest rate n = number of monthly paymentsn = number of monthly payments A = amount of month paymentA = amount of month payment prpr

A = -----------------------A = ----------------------- 1 1 1 - -------------- 1 - -------------- (1 + r) (1 + r)n

ComplexComplex Ration Expression – has complex rational Ration Expression – has complex rational expression in numerator or denominatorexpression in numerator or denominator

Simplifying Simplifying Complex Rational ExpressionComplex Rational Expression

Simplify:Simplify: 1 y 1 y--- + ------ + --- x x x x22

---------------------- 1 x 1 x--- + ------ + --- y y y y22

Find the LCD: x x y y = xFind the LCD: x x y y = x22yy22

MultiplyMultiply all terms by x all terms by x22yy22 / x / x22yy2 2 = 1= 1

(x(x22yy22)1 (x)1 (x22yy22)y xy)y xy2 2 + y+ y3 3 ---------- + ----------- -------------------------- + ----------- ---------------- (x (x22yy22)x (x)x (x22yy22)x)x2 2 x x22yy2 2

----------------------------- = -------------------------------------------------- = --------------------- (x (x22yy22)1 (x)1 (x22yy22)x x)x x22y + xy + x33

---------- + ----------- -------------------------- + ----------- ---------------- (x (x22yy22)y (x)y (x22yy22)y)y2 2 xx22yy2 2

xyxy2 2 + y+ y3 3 y y22(x + y) y(x + y) y22

------------- = -------------- = ------------------ = -------------- = ----- xx22y + xy + x33 x x22(y + x) x(y + x) x22

YourYour TurnTurn

Simplify the following:Simplify the following:

1.1.((x/y) – 1) / ((x((x/y) – 1) / ((x22/y/y22) – 1))) – 1))• Solution: (xy – y2) / (xSolution: (xy – y2) / (x22 – y – y22) = y / (x + y)) = y / (x + y)

2.2.(1/(x + h) – 1/x) / h(1/(x + h) – 1/x) / h• Solution: -1/(x(x + h))Solution: -1/(x(x + h))

SkipSkip

6.4 Division of Polynomial Expressions6.4 Division of Polynomial Expressions 6.5 Synthetic Division6.5 Synthetic Division

6.6 Rational Equations6.6 Rational Equations

Given:Given: Cost (in $1000) of cleaning a lakeCost (in $1000) of cleaning a lake

120x 120xf(x) = ----------f(x) = ---------- 100 – x 100 – xwhere x = % of pollutants to be eliminatedwhere x = % of pollutants to be eliminated

Question:Question: If $80,000 is appropriated for the cleanup, If $80,000 is appropriated for the cleanup,

what % of pollutants can be eliminated?what % of pollutants can be eliminated?

120x120xf(x) = -----------f(x) = ----------- 100 – x 100 – x

Solution:Solution: 200x200x

80 = -----------80 = ----------- 100 – x 100 – x

80(100 – x) = 200x80(100 – x) = 200x8000 – 80x = 200x8000 – 80x = 200x8000 = 280x8000 = 280xx = 28.6(%)x = 28.6(%)

Solving Rational Equation Solving Rational Equation Solve: x + 6 x + 24Solve: x + 6 x + 24

-------- + ---------- = 2 -------- + ---------- = 2 2x 5x 2x 5x

Note: x Note: x ≠ 0≠ 0

x + 6 x + 24 x + 6 x + 24 10x -------- + ---------- = 10x 2 10x -------- + ---------- = 10x 2 2x 5x 2x 5x

5(x + 6) + 2(x + 24) = 20x5(x + 6) + 2(x + 24) = 20x5x + 30 + 2x + 48 = 20x5x + 30 + 2x + 48 = 20x78 = 13x78 = 13xx = 6x = 6

CheckCheck Solve: x + 6 x + 24Solve: x + 6 x + 24

-------- + ---------- = 2 -------- + ---------- = 2 2x 5x 2x 5x

Note: x Note: x ≠ 0≠ 0

6 + 6 6 + 24 ?6 + 6 6 + 24 ?------- + ---------- = 2------- + ---------- = 22(6) 5(6)2(6) 5(6)

12 30 12 30------- + ------- = 2------- + ------- = 2 12 30 12 30

Solving Rational Equation (2)Solving Rational Equation (2) Solve: x 3Solve: x 3

-------- = ---------- + 9 -------- = ---------- + 9 x – 3 x – 3 x – 3 x – 3

Note: x Note: x ≠ 3≠ 3

x 3 x 3 (x – 3) -------- = (x – 3) --------- + 9(x – 3) -------- = (x – 3) --------- + 9 x - 3 x - 3 x - 3 x - 3

x = 3 + (x – 3)9 x = 3 + (x – 3)9 x = 3 + 9x – 27x = 3 + 9x – 27x = -24 + 9xx = -24 + 9x24 = 8x24 = 8xx = 3x = 3

But x cannot be 3. Thus, no solution.But x cannot be 3. Thus, no solution.

Solving Rational Equation (3)Solving Rational Equation (3) SolveSolve

x 9 x 9---- + ----- = 4 ---- + ----- = 4 3 x 3 x

Note: x Note: x ≠ 0≠ 0

x 9 x 9 (3x) ----- + ---- = (3x) 4(3x) ----- + ---- = (3x) 4 3 x 3 x

x(x) + 3(9) = 12x x(x) + 3(9) = 12x xx22 + 27 = 12x + 27 = 12xxx22 – 12x + 27 = 0 – 12x + 27 = 0(x – 3)(x – 9) = 0(x – 3)(x – 9) = 0

x = 3, x = 9x = 3, x = 9

CheckCheck SolveSolve

x 9 x 9---- + ----- = 4 x = 3, x = 9 ---- + ----- = 4 x = 3, x = 9 3 x 3 x

Note: x Note: x ≠ 0≠ 0

3 9 ? 9 9 ? 3 9 ? 9 9 ?--- + --- = 4 ---- + ---- = 4--- + --- = 4 ---- + ---- = 4 3 3 3 9 3 3 3 9

1 + 3 = 4 3 + 1 = 4 1 + 3 = 4 3 + 1 = 4

Your TurnYour Turn Solve:Solve:

x + 4 x + 20x + 4 x + 20-------- + ---------- = 3-------- + ---------- = 3 2x 3x 2x 3x

Solution:Solution: x x ≠≠ 0 0

x + 4 x + 20 x + 4 x + 206x -------- + ---------- = 6x 36x -------- + ---------- = 6x 3 2x 3x 2x 3x

3(x + 4) + 2(x + 20) = 18x3(x + 4) + 2(x + 20) = 18x3x + 12 + 2x + 40 = 18x3x + 12 + 2x + 40 = 18x52 = 13x52 = 13xx = 4x = 4

Your TurnYour Turn Solve:Solve:

2x 6 -282x 6 -28-------- + --------- = -------------------- + --------- = ------------ x – 3 x + 3 x x – 3 x + 3 x22 - 9 - 9

Solution:Solution: x x ≠≠ 3, x 3, x ≠≠ -3 -3

2x 6 -28 2x 6 -28 (x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) -----------(x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) ----------- (x – 3) (x + 3) x (x – 3) (x + 3) x22 - 9 - 9

(x + 3)2x + (x – 3)6 = -28(x + 3)2x + (x – 3)6 = -282x2x22 + 6x + 6x – 18 = -28 + 6x + 6x – 18 = -282x2x22 + 12x + 10 = 0 + 12x + 10 = 0(2x + 2)(x + 5) = 0(2x + 2)(x + 5) = 0x = -1, x = -5 x = -1, x = -5

6.7 Applications (1)6.7 Applications (1)Rate of WorkRate of Work

Suppose: Tom can complete a Web site in 15 Suppose: Tom can complete a Web site in 15 hours, while her friend Amy can complete it in 10 hours, while her friend Amy can complete it in 10 hours. Working together, how many hours will it hours. Working together, how many hours will it take to complete one job?take to complete one job?

Solution:Solution: Hours working together: xHours working together: x Hour with Tom alone: 15Hour with Tom alone: 15 Hours with Amy alone: 10Hours with Amy alone: 10 Tom’s rate: 1/15 per hourTom’s rate: 1/15 per hour Amy’s rate: 1/10 per hourAmy’s rate: 1/10 per hour

Find an equationFind an equation Rate x Time = 1 jobRate x Time = 1 job 1 1 1 1

x ---- + ---- = 1 x ---- + ---- = 1 15 10 15 10 1 1 1 1

(30) x ---- + ------ = (30) · 1(30) x ---- + ------ = (30) · 1 15 10 15 10

2x + 3x = 302x + 3x = 305x = 305x = 30

x = 6 (hours)x = 6 (hours)

Application (2)Application (2)SpeedSpeed

You commute to work a distance of 40 You commute to work a distance of 40 miles and return on the same route at the miles and return on the same route at the end of the day. Your average rate on the end of the day. Your average rate on the return trip is 30 miles per hour faster than return trip is 30 miles per hour faster than your average rate on the outgoing trip. If your average rate on the outgoing trip. If the round trip takes 2 hours, what is your the round trip takes 2 hours, what is your average rate on the outgoing trip to work?average rate on the outgoing trip to work?

SolutionSolution Average speed going (mph): x Average speed going (mph): x Average speed returning: x + 30Average speed returning: x + 30

Find EquationFind Equation distance = speed x timedistance = speed x time time = distance / speedtime = distance / speed (time going) + (time returning) = 2(time going) + (time returning) = 2 40/x + 40/(x + 30) = 240/x + 40/(x + 30) = 2 (x + 30)40 + 40x = 2x(x + 30)(x + 30)40 + 40x = 2x(x + 30) 40x + 1200 + 40x = 2x40x + 1200 + 40x = 2x22 + 60x + 60x 0 = 2x0 = 2x22 - 20x – 1200 - 20x – 1200 0 = x0 = x22 - 10x – 600 - 10x – 600 0 = (x – 30)(x + 20)0 = (x – 30)(x + 20) x = 30; x = 30; x = -20 (has no interpretation)x = -20 (has no interpretation)

Applications (3)Applications (3)Average CostAverage Cost

Cost of running a manufacturing business is Cost of running a manufacturing business is described by the cost function:described by the cost function: C(x) = (fixed cost) + cx, C(x) = (fixed cost) + cx,where where xx is the number of units produced. is the number of units produced.

Average cost for producing one unit is Average cost for producing one unit is described by the average function:described by the average function: (fixed cost) + cx) (fixed cost) + cx) A(x) = ------------------------ A(x) = ------------------------ x x

Suppose a company manufactures robots Suppose a company manufactures robots with a fixed cost of $1,000,000 and $5000 with a fixed cost of $1,000,000 and $5000 per robot.per robot. C(x) = 1,000,000 + 5000x C(x) = 1,000,000 + 5000x

1,000,000 + 5000x 1,000,000 + 5000x A(x) = --------------------------- A(x) = --------------------------- x x

How many robots need to be produced to How many robots need to be produced to bring the average cost down to $5500?bring the average cost down to $5500?

1,000,000 + 5000x1,000,000 + 5000xA(x) = -----------------------------------A(x) = ----------------------------------- x x

5500 = (1,000,000 + 5000x) / x5500 = (1,000,000 + 5000x) / x5500x = 1,000,000 + 5000x5500x = 1,000,000 + 5000x500x = 1,000,000500x = 1,000,000x = 2000 ($)x = 2000 ($)