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Solve for a in the following equation using the quadratic formula:    

Expand through multiplication: (4-x)(x+6)

Solve the equation for t by factoring: 

Make true: 15 ? 7

Madison knows that YGX is a solution with 20% active ingredient for dosages, if the total volume of the standard dose of YGX is 120 ml (cc’s) then:   a) How many ml of active ingredients are in a standard dose of YGX? b) How many standard doses can be made from 375ml of pure (100%) active ingredient? c) The last person working accidentally made 3L of 35% active ingredient solution, how much water does Madison need to add to make it 20% active ingredient?

Graph the linear equation,

Consider a preference relation on bundles of two goods (x,y) defined as follows: (x,y) is at least as good as (x’,y’) whenever the following is true: either x>x’, or x=x’ and y≥y’-1. Is it transitive?

Consider a preference relation on a finite set X={x1,x2,…,xn}. Suppose that it is complete and transitive. Then, the following function represents it: U(xi)= 3log|{j=1,…,n: xi at least as good as xj}|-2020, where |.| represents the cardinality of a set.

A marketing firm is interested in learning about millennials’ preferences on learning experiences. In order to do this, they survey students at a university. They ask them six questions as follows. They consider the following leaning experiences: a traditional class, a flipped course (half of the content is delivered online), and an online class. For each pair of these alternatives, say a and b, they ask the students “is a at least as good as b.” In order to make this operational, they provide the students with a picture in which the three alternatives are depicted as circles with their description next to it. The students are asked to draw arrows between the alternatives whenever the answer to a question is affirmative. That is, if a student finds alternative a is at least as good as alternative b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question is a at least as good as a was asked; assume that the answer to each of these trivial questions is affirmative.) Suppose that a student’s answer has no arrows.

Julia has preferences on bundles of two commodities. The following figure shows one of her indifference curves. This curve is actually a line that passes through bundles C and D. Suppose that all other indifference curves of Julia are lines with the same slope as the indifference curve shown. What is Julia’s preferred bundle from all the bundles in the segment of line that joins bundles A and B?