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U - union ^ - Intersection AUBUC = A + B + C - [A^C] - [A^B] - [B^C] + A^B^C

We have all the values given in the brackets. We need A^B^C

1st clause (if we draw a diagram its easier) It gives A^B = 16 out of which 9 are in C. i.e. indirectly its giving (A^B)^C and thats what we want A^B^C = 9 (** imp)

2nd clause gives A , B and C values but as we do not know AUBUC (according to formula) we cannot calculate A^B^C.

Hence the answer is A

3. Formulating using the given formula

20 S.P. - 20 C.P = Gross Profit.

1st clause says that Gross Profit = 2400 if S.P = 2 C.P ( ** imp step) i.e (20*2 *C.P) - 20*C.P = 2400 Hence C.P = 120 and S.P = 240 to get a profit of 2400. However we cannot come to any consensus about the Gross Profit really earned as we cannot derive anything about current C.P or S.P.

2nd Clause

Suppose actual S.P = C.P + x According to conditions 20 * ( C.P. + x + 2) - 20 * C.P = 440 Therefore, 20x + 40 = 440 x = 20 Therefore original S.P = C.P. + 20 And Gross Profit = 20 * (C.P+ 20) - 20 * C.P G.P = 400 $

You can directly take 20 (S.P. + 2) - 20 C.P = 440 that gives (S.P. - C.P) = 20 That is the profit for single item We have 20 items hence the Gross Profit = 20 * 20 = 400 $.

In fact more suitably 20 S.P - 20 .C.P = 400 $ and here LHS is what we have assumed as our formula. I hope this is a better alternative to understand.

Simplify question: taking the square root of (x-5)² is equal to consider solutions for the absolute value of x-5

|x-5| could equal either x-5 or 5-x, in other words we need to determine whether x is positive or negative to answer the question

Statement 1: sufficient as we are told that -x multiplied by the absolue value of x is negative. that means that on of -x or |x| is negative and the other is positive. As |x| HAS to be positive then -x is negative. We therefore know that x is positive and can answer the question with NO.

Statement 2: I cannot understand here why this one is sufficient. If we simplify this statement we get x < 5 which i believe is not sufficient alone.

Simplify question: taking the square root of (x-5)² is equal to consider solutions for the absolute value of x-5

|x-5| could equal either x-5 or 5-x, in other words we need to determine whether x is positive or negative to answer the question

Statement 1: sufficient as we are told that -x multiplied by the absolue value of x is negative. that means that on of -x or |x| is negative and the other is positive. As |x| HAS to be positive then -x is negative. We therefore know that x is positive and can answer the question with NO.

Statement 2: I cannot understand here why this one is sufficient. If we simplify this statement we get x < 5 which i believe is not sufficient alone.

Can someone help???

Thx!

Is \(\sqrt{(x-5)^2}=5-x\)?

Remember: \(\sqrt{x^2}=|x|\).

So "is \(\sqrt{(x-5)^2}=5-x\)?" becomes: is \(|x-5|=5-x\)?

Now LHS=sqrt=absolute value>=0, (\(LHS\geq{0}\), as absolute value is never negative), hence RHS also must be more than zero --> \(RHS=5-x\geq{0}\) --> \(x\leq{5}\).

For \(x\leq{5}\) --> \(\{RHS=|x-5|=5-x\}=\{LHS=5-x\}\). So we have that if \(x\leq{5}\), then \(|x-5|=5-x\) is true.

Basically question asks is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\). Sufficient.

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