GMAT Official Guide Quantitative Review 2019_ Book

Algebra

7-x

Given that 4 < , it follows that 12 < 7 -x.

Then, 5 < -x or, equivalently, x <—5.

I. If4<

7-x

, then x < -5. If 5 < x were true

3

then, by combining 5 < x and x < -5, it

would follow that 5 < -5, which cannot be

true. Therefore, it is not the case that, if

7-x

4 < , then Statement I must be true. In

fact, Statement I is never true.

7-x

II. If 4 < , then x < -5, and it follows that

3

x + 3 < -2. Since -2 < 0, then x + 3 < 0 and

Ix+ 31 =-(x+3).If x+ 3 < -2, then-(x+ 3) >2

and by substitution, Ix+ 3 I > 2. Therefore,

Statement II must be true for every value of

x such that x < -5. Therefore, Statement II

7-x

must be true if 4 <—.

7-x

III. If 4 < , then x < -5 and x + 5 < 0. But if

3

x + 5 < 0, then it follows that—(x + 5) > 0

and so -(x + 5) is positive. Therefore

7-x

.

3

Statement III must be true if 4 <

1h e correct answer 1s D.

PS08654

164. A certain right triangle has sides of length x, y, and z,

(B)

(C)

(D)

(E)

where x < y < z. If the area of this triangular region is 1,

which of the following indicates all of the possible

values of y?

(A) y > ./2_

求

2

<Y<.fi.

一迈<Y<—岛

3 2

—岛<Y<一^5

4 3

Y<— 岛

4

4.5 Answer Explanations

Geometry; Algebra

Since x, y, and z are the side lengths of a right

triangle and x < y < z, it follows that x and y are

the lengths of the legs of the triangle and so the

area of the triangle is -^1 xy. But, it is given that

2

the area is 1 and so—^1 xy = 1. Then, xy = 2 and

2 2

y=—. Under the assumption that x,y, and z

X

are all positive since they are the side lengths of

a triangle, x < y implies—1 1 > -and then -2 2 >一．

But, y =-,so b^2 y subst1tut10n,. X. y y >-^2 ,which X y

X J

implies that y2 > 2 since y is positive. Thus, y >五

Alternatively, if x < 立ndy<拉thenxy < 2.

If x > J2 and y > 拉，thenxy > 2. But, xy = 2 so

one of x or y must be less than迈andthe other

must be greater than J2. Since x < y, it follows

that x < J2 < y and y > J2.

1h e correct answer 1s A.

PS14397

165. On a certain day, a bakery produced a batch of rolls at
     a total production cost of $300. On that day^4
     , 一ofthe
     5
     rolls in the batch were sold, each at a price that was
     50 percent greater than the average (arithmetic mean)
     production cost per roll. The remaining rolls in the
     batch were sold the next day, each at a price that was
     20 percent less than the price of the day before. What
     was the bakery's profit on this batch of rolls?

(Al $150

(8) $144

(Cl $132

(D) $108

(El $90