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Lalph Rauren was famous for discovering potential models as he walked the streets of New York. One day he spotted four girls who later became top models: Amy, Mary, Kay, and Beth. One was discovered walking through an apartment building, one in a coffee shop, one in a shopping mall, and one in the library. You know the following.

a.If Kay was not discovered in the library, then Amy was discovered in the apartment building.

b.Beth was discovered in the library, unless Mary was discovered in the coffee shop.

c.If Amy was discovered in the apartment building, then Mary was discovered in the shopping mall.

d.If Beth was not discovered in the library, then neither was Kay.

If you have enough information from the statements above, match the girl with the location in which she was discovered and explain your reasoning. If you cannot make the match, explain why.
No one has answered this question yet.

Can it even be solved?

We can solve this problem in many ways, a truth table, or a logic tree. The latter seems more convenient, so it will be used.

First let's use some definitions of symbols.

For the names,
A=Amy
B=Beth
K=Kay
M=Mary

For the locations,
A=Apartment building
C=Coffee Shop
S=Shopping Mall
L=Library

Now the operators:
K=L means Kay was discovered in the library (etc.)
K≠L means Kay was not discovered in the library.
-> means "then", or "it follows".
and finally,
&xor; means exclusive or, exactly one of the two statements is true.

In logic, if the condition of a conditional statement is true, "it follows" that the following statement
is true. If the condition is NOT satisfied, we do not know if the following statement is true or not, i.e.
a->b
if a is true, we know that b is true
if a is not true, we don't know if b is true.

"If it rains, I stay home."
If it doesn't rain, I may stay home, or I may go out.

Now let's get started.

We will translate the four given statements:
1. K≠L ->A=A
2. B=L &xor; M=S
3. A=A -> M=S
4. B≠L -> K≠L

We have two cases, either (A) K was found in the Library, or (B) she was not.

case (A)
K=L
A=A (from 1.)
B≠L (since K=L)
But from 4, B≠L -> K≠L

Case (B)
K≠L
A=A (from 1, K≠L -> A=A)
M=S (from 3 A=A -> M=S)
B=L
(if(B≠L, then K≠L, then nobody was found in the library; => B=L)
K=C (by elimination)

Conclusion:
K=C,B=L,M=S,A=A
If you have been following the logic, you would be able to translate the conclusion.

Solve the following inequality. Then place the correct answer in the box provided. Answer in terms of an improper fraction.

3y + 5 >10