Solve the equation algebraicly for the exact vaule of x check your work : the equation is 6 - 2/4 (x + 5) = 4x

x = 7/9

Step-by-step explanation:

Solve for x:

(-x - 5)/2 + 6 = 4 x

Put each term in (-x - 5)/2 + 6 over the common denominator 2: (-x - 5)/2 + 6 = 12/2 + (-x - 5)/2:

12/2 + (-x - 5)/2 = 4 x

12/2 + (-x - 5)/2 = ((-x - 5) + 12)/2:

(-x - 5 + 12)/2 = 4 x

Grouping like terms, -x - 5 + 12 = (12 - 5) - x:

((12 - 5) - x)/2 = 4 x

12 - 5 = 7:

(7 - x)/2 = 4 x

Multiply both sides by 2:

(2 (7 - x))/2 = 2×4 x

(2 (7 - x))/2 = 2/2×(7 - x) = 7 - x:

7 - x = 2×4 x

2×4 = 8:

7 - x = 8 x

Subtract 8 x from both sides:

7 + (-x - 8 x) = 8 x - 8 x

-x - 8 x = -9 x:

-9 x + 7 = 8 x - 8 x

8 x - 8 x = 0:

7 - 9 x = 0

Subtract 7 from both sides:

(7 - 7) - 9 x = -7

7 - 7 = 0:

-9 x = -7

Divide both sides of -9 x = -7 by -9:

(-9 x)/(-9) = (-7)/(-9)

(-9)/(-9) = 1:

x = (-7)/(-9)

Multiply numerator and denominator of (-7)/(-9) by -1:

 x = 7/9

step-by-step explanation:

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).