What is the Best And Simple Approach

[gmataspirant]

Could anyone explain the best approach to solve the following question.
I am sure that you can solve it by plugging some values, just looking for innovative ways to solve this question.

1. If 2<x<5 and 3<y<10 which of the following expresses the possible range of values for x-y?

A) -1 < x-y < 5
B) -1 < x-y < 2
C) -5 < x-y < 2
D) -8 < x-y < 2
E) -8 < x-y < -1

Your detailed explainations are welcome.

[Ian Stewart]

Could anyone explain the best approach to solve the following question.
I am sure that you can solve it by plugging some values, just looking for innovative ways to solve this question.

1. If 2<x<5 and 3<y<10 which of the following expresses the possible range of values for x-y?

A) -1 < x-y < 5
B) -1 < x-y < 2
C) -5 < x-y < 2
D) -8 < x-y < 2
E) -8 < x-y < -1

Your detailed explainations are welcome.

When will x-y be largest? When x is largest and y is smallest. The max value will be 5-3 = 2. When will x-y be smallest? When x is smallest and y is largest. The min value will be 2-10 = -8. The inequalities are strict, though, so x-y can never strictly equal these two boundary values, but it can come as close as you like. Thus, -8 < x-y < 2.

In general, with these questions, if x and y are combined by addition, subtraction or multiplication, you only need to do four things- combine:

max, max
max, min
min, max
min, min

you'll find all the extreme values. So if a question asks for the range of values of x*y if

-4 < x < 3
-5 < y < 6

you don't need to think much; just calculate 3*6 = 18; (-4)*(-5) = 20; (-4)*(6) = -24; (-5)*3 = -15. The max is 20, the min is -24.

[AleksandrM]

Ian,

I have a quetion. Why are you choosing values for x and y that the problem states fall outside of the range of possible values of x and y?

For example, you choose 5 as the largest value of x, when the problem states that x is less than 5 and greater than 2. You did the same for y.

Thanks.

[reetchatha]

Is D the rite answer?

[Ian Stewart]

Ian,

I have a quetion. Why are you choosing values for x and y that the problem states fall outside of the range of possible values of x and y?

For example, you choose 5 as the largest value of x, when the problem states that x is less than 5 and greater than 2. You did the same for y.

Thanks.

Alexsander,

It's certainly true that x cannot equal 5 here, but x can be as close to 5 as you like (provided it's less than 5). In calculus-speak (which you don't need for the GMAT!), we say x can be infinitesimally close to 5. Imagine it as being 4.9999... with a long string of 9s if you like, but that just makes calculation difficult!

Since 5 acts as a sort of 'ceiling' for x, we can use it as the maximum value, as long as we bear in mind that x can never quite equal 5. Say
0 < x < 5
0 < y < 3
and we want the range of possible values of x+y. We want to use the boundary values for each-- 0 and 5 for x, 0 and 3 for y--, but if we want the range of values for x+y we need to acknowledge that 0 < x+y < 8; it's impossible for x+y to exactly be equal to 0 or equal to 8, but we can get as close as we want.

[AleksandrM]

Kool. Thanks.

[gmataspirant]

max, max
max, min
min, max
min, min

you'll find all the extreme values. So if a question asks for the range of values of x*y if

-4 < x < 3
-5 < y < 6

you don't need to think much; just calculate 3*6 = 18; (-4)*(-5) = 20; (-4)*(6) = -24; (-5)*3 = -15. The max is 20, the min is -24.

Excellent Ian, Great Explanation.

[durgesh79]

another way of doing it.

When an inequalities is multipied by a -ve number, the signs are reversed

we have If 2<x<5 and 3<y<10> -y > -10
can be written as -10 < -y < -3

add 2<x<5 and -10<-y<-3

8 < x-y < 2 . answer