Four-fifths of my current age is greater than three-quarters of my age one year from now. four-fifths of my current age is also greater than five-sixths of my age one year ago. given that my age is an integer, what are all possible values for my age? explain how you determined your answer, showing all steps along the way.

Let current age be x, next we construct the inequalities for the age:
Four-fifths of my current age is greater than three-quarters of my age one year from now.
thus when we add 1 year to 3/4 of our current age and set the inequality
4x/5>3x/4+1i

Four-fifths of my current age is also greater than five-sixths of my age one year ago.
Thus when we subtract 1 from 5x/6 and set the inequality we get
4x/5>5x/6-1ii
solving the inequalities we obtain:
4x/5>3x/4+1
x/20>1
hence multiplying both sides by 20 we obtain:
x>20

also

4x/5>5x/6-1
4x/5-5x/5>-1
-x/30>-1
multiplying both sides by 30 we get:
-x>-30
thus
x<30
therefore the possible values of my age will lie in the interval:
20<x<30
Thus our age is in the interval
(20,30)

  b.   the average rate of change of f(x) is less than the average rate of change of g(x).

step-by-step explanation:

since the length of the interval is the same in each case, we only need to compare the change in y-value to see which is greater.

f(-2) = -3f(1) = 6 . . . estimated from f(x) = (x+2)^2-3, which matches points on the graph (-1,-2) and (0, 1)

so the change in y-value for function f is (6 )) = 9.

the table tells us

g(-2) = 1g(1) = 31

so the change in y-value for function g is (31 -1) = 30.

the change in y-value of f(x) on the interval is less than the change in y-value of g(x) on the same interval, so the average rate of change of f(x) is less than the average rate of change of g(x).

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