Let -? and ? denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on R U {?} U {-?} as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for t ? R define

t? = { -? if t<0 , 0 if t=0, ? if t>0

t(-?) = { ? if t<0, 0 if t=0, -? if t>0

t+? = ?+t=?, t+(-?)=(-?)+t=-?, ?+?=?, (-?)+(-?)=-?, ?+(-?)=0

IsR U {?} U {-?} a vector space over R? Please Explain.

Find dy/dx using implicit differentiation ln(20+e^xy)=y

The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. the expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $95 and a standard deviation of $20. if the management wants to give free gifts to at most 10% of the customers, what should the amount be above which a customer would receive a free gift?

What would be the reasons for lines 3 and 4? a. addition property; subtraction property b. addition property; solve c. substitution property; subtraction property d. substitution property; prove