# It costs d dollars to ship a package that weighs p pounds if and only if 6<=p<=10. If Alan's package weighs p pounds and costs d dollars to ship, which of the following must be true?...

It costs d dollars to ship a package that weighs p pounds if and only if 6<=p<=10. If Alan's package weighs p pounds and costs d dollars to ship, which of the following must be true?

A)p-6<=2

B)p-10<=2

C)p-8<=2

D)p-6<=10

E)p+10<=6

Answer is C but do not understand

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### 1 Answer

Let **P** be the statement "it costs d dollars to ship a package that weighs p pounds.

Let **Q **be the statement "`6 <= p <= 10.`

Hence, what we have here is `P hArr Q.`

The "if and only if" phrase is biconditional, and it means "both a conditional and its converse". We can divide this into two statements:

- If the shipping cost is d dollars, then the weight of the package is between 6 and 10 (inclusive) pounds.
- If the package is between 6 and 10 (inclusive) pounds, then the shipping cost is d dollars.

Now, we know that Allan's package weighs p pounds and d dollars. This satisfies the condition (in the first statement above) so the weight of his package must be `6 <= p <= 10.`

This statement, on the other hand, is "p is between 6 and 10". Therefore, we only need to find, among the choices, something that is not contradictory to that statement.

Notice that letter C, `p-8<=2` is equivalent to saying `p<= 10` .

The other letters are translated, respectively to:

- `p <= 8`
- `p <= 12`
- `p <= 10`
- `p<=16`
- `p<=-4`

Note that B, D, E are easily eliminated among the choices. B and D states the possibility of having a weight greater than 10 which is clearly restricted; while E is merely absurd. A, on the other hand, might be wrong as it states that it is not possible to have a weight of 9 pounds, when in fact the original statement doesn't forbid us to have a 9-pound package. Hence, the answer is C. Strictly speaking, however, we still need the lower bound. For instance, a 2-pound package satisfies C but not the original statement. Another, note, letter A (technically speaking) is not wrong as a package with weight less than 8 pounds still satisfies the given condition. Nonetheless, the best answer to this is again, letter C (as explained above).

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