6.5 – Solving Equations w/ Rational Expressions

6.5 – Solving Equations w/ Rational Expressions

5 16 1x 4 14 5 20x

4 14 5

20 2020

20x

5x

LCD: 20

5 16 1x

5 15x

3x

44 11

LCD:

2 3 3 3 2x x 2

2 3 23 3 9x x x

2 3 2

3 3 3 3x x x x

3 3x x

3 3 23x

x x

2 3 3 3 2x x

2 6 3 9 2x x

5 3 2x

5 5x 1x

3 33

3x

x x

3

3 33 2x x

x x


LCD: 6x5 3 33 2 2x

6 53

x

2 5 3 3 3 3x x 10 9 9x x

9x

362x

x 2

6 3x

10 9 9x x


LCD: x+36 2 23 3

xxx x

3x x

2 3x x

2 12 0x x 3x

3 0 4 0x x 3x

3 63x

x

233

x xx

3 2x

6 2x 2 6x 2 3x x 6 4 6x

0 4x 4x


LCD:

2

5 11 1 122 7 10 5

xx x x x

5 11 1 12

2 2 5 5x

x x x x

2 5x x

2 5 52x

x x

5 5 11 1 12 2x x x

5 25 11 1 12 24x x x

5 25 23x x

6 48x 8x

12 5

2 1 15 xx

xx

x

2 5 12

5x

xx


LCD: abx1 1 1a b x

1abxa

bx

bx ab ax

bx

bxb x

Solve for a

1b

abx 1x

abx

ax ab

a b x

a


Problems about NumbersIf one more than three times a number is divided by the number, the result is four thirds. Find the number.

3x

3 3 1x xx

3 3 1 4x x

9 3 4x x

5 3x 35

x

LCD = 3x1x

43

3

3 4x

9 4 3x x

6.6 – Rational Equations and Problem Solving

Problems about Work Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can sort a batch in 3 hours. If they work together, how fast can they sort one batch?

Time to sort one batch (hours)

Fraction of the job completed in one hour

Ryan

Mike

Together

12131x

2

3

x


Problems about Work Time to sort

one batch (hours)


Ryan

Mike

Together

12131x

2

3

x

12 6 1

2x

3x 5 6x 65

x hrs.

LCD =13

1x

6x 3

6 1x 16x

x

2x 6115


James and Andy mow lawns. It takes James 2 hours to mow an acre while it takes Andy 8 hours. How long will it take them to mow one acre if they work together?

Time to mow one acre (hours)


James

Andy

Together

2

8

x

12181x


Time to mow one acre (hours)


James

Andy

Together

2

8

x

12181x

LCD:12

8 12

x

4x 5 8x 85

x

hrs.

18

1x

8x 8

8 1x 18x

x

x 8315


A sump pump can pump water out of a basement in twelve hours. If a second pump is added, the job would only take six and two-thirds hours. How long would it take the second pump to do the job alone?

112

1x

263

1203

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

203


112

1x

263

1203

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

112

1 112 x

203

1x

1203

320


LCD:

60 112

x

5x

60 4xhrs. 15x

1 1 312 20x

60x

160x

x 0

60 32

x

60 9x

5x60 9x


Distance, Rate and Time Problems If you drive at a constant speed of 65 miles per hour and you travel for 2 hours, how far did you drive?

d r t

65mileshour

2 hours 130 miles

d tr

d rt


A car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles.

Rate Time Distance

Motor-cycle

Car

x

x + 15

450 mi

600 mi

t

t

rdt

x450

15600x


Rate Time Distance

Motor-cycle

Car

x

x + 15

450 mi

600 mi

t

t

rdt

x450

15600x

x450

15

600x

LCD: x(x + 15)

15600450

xx

x(x + 15) x(x + 15)


15600450

xx

x(x + 15) x(x + 15)

xx 60045015

xx 60015450450

x15015450

x

15015450

x45

45x mphMotorcycle

6015 x mphCar


Rate Time Distance

UpStream

DownStream

A boat can travel twenty-two miles upstream in the same amount of time it can travel forty-two miles downstream. The speed of the current is five miles per hour. What is the speed of the boat in still water? boat speed = x

x - 5

x + 5

22 mi

42 mi

t

t

rdt

522x

542x


Rate Time Distance

UpStream

DownStream

boat speed = x

x - 5

x + 5

22 mi

42 mi

t

t

rdt

522x

542x

522x

5

42x

LCD: (x – 5)(x + 5)

542

522

xx(x – 5)(x + 5) (x – 5)(x + 5)


542225 xx

2104211022 xx

xx 2242210110

x20

320

x1616 mph

Boat Speed

542

522

xx(x – 5)(x + 5) (x – 5)(x + 5)

x20320


Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such that

6.7 – Variation and Problem Solving

The number k is called the constant of variation or the constant of proportionality

.kxy

Direct Variation

kxy 824 k

k824


Suppose y varies directly as x. If y is 24 when x is 8, find the constant of variation (k) and the direct variation equation.

3k

xy 3direct variation equation

constant of variation

xy

39

515

927

1339

kwd 567 k

k567

6.7 – Variation and Problem SolvingHooke’s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 56-pound weight stretches a spring 7 inches, find the distance that an 85-pound weight stretches the spring. Round to tenths.

81

k

xy31

direct variation equation


8531

y

6.10y inches

Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that


The number k is called the constant of variation or the constant of proportionality.

.xk

y

Inverse Variation

xk

y

36

k

k18


Suppose y varies inversely as x. If y is 6 when x is 3, find the constant of variation (k) and the inverse variation equation.

xy

18



xy

36

92

101.8

181

tk

r

430

k

k120

6.7 – Variation and Problem SolvingThe speed r at which one needs to drive in order to travel a constant distance is inversely proportional to the time t. A fixed distance can be driven in 4 hours at a rate of 30 mph. Find the rate needed to drive the same distance in 5 hours.

xr

120



5120

r

24r mph

Joint Variation6.7 – Variation and Problem Solving

If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional, to the other variables.Therefore, if , then the number k is the constant of variation or the constant of proportionality.

𝑧=𝑘𝑥𝑦

z varies jointly as x and y. x = 3 and y = 2 when z = 12. Find z when x = 4 and y = 5.

12=𝑘 (3 ) (2 ) 2=𝑘 𝑧=2 𝑥𝑦𝑧=2 (4 ) (5 ) 𝑧=40

Joint Variation6.7 – Variation and Problem Solving

𝑉=𝑘h𝑟 2

V varies jointly as h and . V = 402.12 cubic inches, h = 8 inches and r = 4 inches. Find V when h = 10 and r = 2.

402.12=𝑘 (8 ) ( 4 )2 3.142=𝑘𝑉=3.142h𝑟 2 𝑖𝑛3𝑉=125.68

The volume of a can varies jointly as the height of the can and the square of its radius. A can with an 8 inch height and 4 inch radius has a volume of 402.12 cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch height?

𝑉=3.142 (10 )22

Additional Problems

LCD: 15

5 2 3 1 1x x 2 1 13 5 15x x

15 152 1 13 5 15

15x x

5 2x

5 10 3 3 1x x

2 13 1x

2 12x

6x 13 x 11


LCD: x

22 6 7x x x 62 7xx

62 7xx

x x x x

2x

20 5 6x x

1 0 6 0x x

1 6x x

0 1 6x x

6 2x 7x


LCD:5 5 31 1x

x x

51

1x xx

5 5 3 3x x

2 2x 1x

1x

1 51x

x

1 3x

5 3 5 3x x

Not a solution as equations is undefined at x = 1.


Problems about Numbers

The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number.

2x

62x

3 2x x 3 2x x

1x

LCD = 613

6x

3

6 1 6

6x

2 2x


7.1 – RadicalsRadical Expressions

Finding a root of a number is the inverse operation of raising a number to a power.

This symbol is the radical or the radical sign

n aindex radical sign

radicand

The expression under the radical sign is the radicand.

The index defines the root to be taken.

Radical Expressions

The symbol represents the negative root of a number.

The above symbol represents the positive or principal root of a number.

7.1 – Radicals

Square Roots

If a is a positive number, then

a is the positive square root of a and

100

a is the negative square root of a.

A square root of any positive number has two roots – one is positive and the other is negative.

Examples:

10

2549

57

0.81 0.9

36 6

9 non-real #

8x 4x

7.1 – Radicals

RdicalsCube Roots

3 27

A cube root of any positive number is positive.

Examples:

354

312564

3 8 2

A cube root of any negative number is negative.

3 a

3 3x x 3 12x 4x

7.1 – Radicals

nth Roots

An nth root of any number a is a number whose nth power is a.

Examples:

2

4 81 3

4 16

5 32 2

43 81

42 16

52 32

7.1 – Radicals

nth Roots

4 16

An nth root of any number a is a number whose nth power is a.

Examples:

15 1

Non-real number

6 1 Non-real number

3 27 3

7.1 – Radicals

15023 2 xxxx2

xx 62 2 x6 15

6

186 x

3

33

x

3152 2

xx

6.4 – Synthetic Division

𝑥−3=0𝑥=3

3 −2 0 15

−2−6−6−18−3

−2 𝑥−6− 3𝑥−3

6.4 – Synthetic Division

2 𝑥−1=0𝑥=1/2

1/2 8 2 −7

8463−4

8 𝑥 +6− 4𝑥−3

12728 2

xxx

12 x

12434x

x

Synthetic division will only work with linear factors with an one as the x-coefficient.

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6.5 – Solving Equations w/ Rational Expressions