Let *y be the operation given by *y = 4/y – y. Which of the following

Question

Let *y be the operation given by *y = 4/y – y. Which of the following statements are true?

I. If 0 < y, then *y is negative.
II. If 0 < y < z, then *y > *z.
III. If 0 < y then y(*y) is less than 5.

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

Luckisnoexcuse · Answer

I) let y=2 then *y = 2-2=0...not true
II) let y = 2 and z = 4 then *y=0 and *z=-3; let y=1/50 and z = 1 then *y=~200 and *z=3...true (any value above 4 gives negative *y)
III) let y=1 then y(*y)=3; y=1/50 then y(*y) =~4; y=100 then y*y is -ve....true

D

ScottTargetTestPrep · Answer

Let’s analyze each answer choice: I. If 0 < y, then *y is negative. We can let y = 1, and we have: *1 = 4/1 - 1 = 4 - 1 = 3 We see that I is not true since *1 is not negative. II. If 0 < y < z, then *y > *z. We can let y = 1 and z = 2. Recall that *1 = 3 and: *2 = 4/2 - 2 = 0 We see that *1 is greater than *2. However, let’s prove the statement algebraically. We are given that y < z, so we have: 1/y > 1/z (recall that when we reciprocate two positive quantities, we switch the inequality sign) and -y > -z (recall that when we multiply two quantities by a negative number, we switch the inequality sign) Multiplying the first inequality by 4 and adding the result to the second inequality, we have: 4/y - y > 4/z - z *y > *z This proves that II is true. III. If 0 < y, then y(*y) is less than 5. Notice that y(*y) = y(4/y - y) = 4 - y^2. Since y^2 is a nonnegative quantity, 4 - y^2 will be no more than 4, which means it’s always less than 5. So III is true. Answer: D

FreshStove · Answer

I am still not able to understand the answer. For each of the three cases I, II and III, I think there are cases where the mentioned is not true. So would we still consider it? Like if we consider y and z as fractions?

Bunuel · Answer

Let *y be the operation given by *y = 4/y – y. Which of the following statements are true? I. If 0 < y, then *y is negative. II. If 0 < y < z, then *y > *z. III. If 0 < y then y(*y) is less than 5. (A) I only (B) II only (C) III only (D) II and III (E) I, II, and III Given: *y = 4/y - y. I. If 0 < y, then *y is negative. If y = 1, then *y = 4/1 - 1 = 3, which is positive. So this statement is not true. II. If 0 < y < z, then *y > *z. For positive y, as y gets larger, 4/y gets smaller and -y also gets smaller. So *y decreases as y increases. Thus, if 0 < y < z, then *y > *z. So this statement is true . III. If 0 < y then y(*y) is less than 5. y(*y) = y(4/y - y) = 4 - y^2 Since y > 0, y^2 > 0, so: 4 - y^2 < 4 < 5 So this statement is true . Answer: D.

egmat · Answer

Hi FreshStove,

You've spotted the exact thing that decides these problems, so let's nail it down. When a statement says  "If 0 < y, then ..." , it is only counted as  true  when it holds for  every  allowed value of y - fractions included. So there are two different jobs:

- To call a statement  false , you need just  one  counterexample.
- To call a statement  true , it must survive  all  values - you can't find even one exception.

That's why the posts above dismissed  I  so fast:  *1 = 4/1 − 1 = 3 , which is positive. One counterexample is enough - I is out.

Now to your real worry:  can fractions break II or III?  Let's actually test them instead of guessing.

Statement III:  rewrite it the way Scott did -  y(*y) = 4 − y2 . Try a fraction,  y =  1/2  : that's   4  −  1/4  =  3.75  , still under  5 . In fact, since y is positive,  y2  is positive, so you're always subtracting something from  4 . The result can never even reach  4 , let alone  5 .  No fraction can break it.

Statement II:  pick  y =  1/2   and  z =  1  . Then  *(1/2) =  8  −  1/2  =  7.5   and  *1 =  3  . The smaller input gave the bigger output - exactly what II claims. This always happens: as y grows,  4/y  shrinks  and−y  shrinks, so  *y  only ever goes down.  Fractions follow the same rule.

So the reason the answer stays  D  is simple: I has a counterexample, but II and III have  none  - not with whole numbers, not with fractions. Try to build a fraction counterexample to III yourself; watch how  4 − y2  refuses to reach  5 . That's the whole idea.

Answer: D

Let y be the operation given by y = 4/y – y. Which of the following

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