Let S = {A1, A2, A3, A4, A5} be a collection of five subsets of the set A = {−5, −4, . . . , 4, 5}, where A1 = {x ∈ A : 1 < x2 < 10} A2 = {x ∈ A : (x + 2)(x − 4) > 0} A3 = {x ∈ A : |x + 2| + |x − 3| ≤ 5} A4 = {x ∈ A : 1 x2+1 > 2 5 } A5 = {x ∈ A : sin πx 4 = 0}. A relation R is defined on S by Ai R Aj (1 ≤ i, j ≤ 5) if Ai and Aj are numerically equivalent. According to Theorem 11.1, R is an equivalence relation. Determine the distinct equivalence classes for this equivalence relation.

a

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

the formula of a slope:

we have the points (-2, 7) and (2, 3). substitute:

27/4x^6 y^8

step-by-step explanation:

4(3x^2y^4)^3 (2x^3y^5)^4

=

108x^6 y^12 /16x^12 y^20

=

108 /16x^6 y^8

=

27 /4x^6 y^8