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(1) w > 2. Subtract \(w > 2\) from \(wz < 2\) (we can do it as the signs are in opposite direction):
\(wz-w<2-2\) --> \(w(z-1)<0\), as w is positive (given w>2), then the product to be negative \(z-1\) must be negative --> \(z-1<0\) --> \(z<1\). Sufficient.
(2) z < 2. If \(z=1.5>1\) and \(w=0\) (wz<2), then the answer is YES but if \(z=0<1\) and \(w=0\) (wz<0), then the answer is NO. Not sufficient.
Here the simple way to conclude that statement 1 is sufficient is because w is always positive and greater than 2 for w=2 which it cant be => z=1 so as Z increases z will decrease => Z<1
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Statement 1: w > 2 This means that w is POSITIVE, which means we can divide both sides of the given inequality, wz < 2, by w. We get: z < 2/w First, since w is POSITIVE, we know that 2/w is POSITIVE, which means z is POSITIVE Second, since w > 2, we know that 2/w will be less than 1, since the denominator is greater than the numerator So, we can write 2/w < 1 Since z < 2/w, we can write: z < 2/w < 1 This means we can conclude that z < 1 Since we can answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: z < 2 There are several values of w and z that satisfy statement 2 (and the given info). Here are two: Case a: w = 1 and z = 0. Notice that this satisfies the given info that wz < 2. In this case z < 1 Case b: w = 1 and z = 1.5. Notice that this satisfies the given info that wz < 2. In this case z > 1 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
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