Problem Solving

If −3<x<5 and −7<y<9, which of the following represents the range of all possible values of y−x?
a. −4<y−x<4
b. −2<y−x<4
c. −12<y−x<4
d. −12<y−x<12
e. 4<y−x<12

In the solution provided for correct ans D,the max value of y is taken as 9 whereas the min value of x is taken as -3.Is this a mistake in the question or we can ignore < and consider it as <= ?

-7<y<9 ..... 1
-3<x<5 ..... 2
------------------
-4<y-x<4

So the correct answer in my opinion should be a.
They have asked for range and not the maxim or minimum value.

There seems to be no mistake found in q. [spoiler]"D"[/spoiler] is the only option which gives you range. All remaining options wont give proper result. You must be agree High(Y) is 8 and min(X) is -2 So (Y-X)= 10 . Which eliminates options A, B and C.
Now for lower bound: min(Y)=-6 and max(X)=4 So (Y-X) =-10
which eliminates option E.
Now you left with only one option, which is your answer.

Thanks,
beatthe800

shreerajp99 wrote:If −3<x<5 and −7<y<9, which of the following represents the range of all possible values of y−x?
a. −4<y−x<4
b. −2<y−x<4
c. −12<y−x<4
d. −12<y−x<12
e. 4<y−x<12

To determine the lower limit and the upper limit of y-x, calculate y-x using every combination of endpoints (-7 and 9 for y, -3 and 5 for x):
-7-(-3) = -4.
-7-5 = -12.
9-(-3) = 12.
9-5 = 4.

The LOWER limit is the LEAST of the results above: -12.
The UPPER limit is the GREATEST of the results above: 12.
Thus:
-12 < y-x < 12.

The correct answer is D.

An alternate approach is to compare the answer choices and use process of elimination.

A, B and C indicate that y-x < 4.
D and E indicate that y-x < 12.
Try to plug in a combination of values such that y-x is BETWEEN 4 and 12.
If y=8 and x=-2, then y-x = 8-(-2) = 10.
Since it's possible that y-x > 4, eliminate A, B and C.

D indicates that y-x > -12.
E indicates that y-x > 4.
Try to plug in a combination of values such that y-x is BETWEEN -12 and 4.
If y=-6 and x=4, then y-x = -6-4 = -10.
Since it's possible that y-x < 4, eliminate E.

The correct answer is D.

Hey shreerajp99 try this Quick and easy approach (without substitutions)

Given : -3<x<5 and -7<y<9
To find : y - x

Solution :
y-x = y + (-x)
Now we know the range of x is -3<x<5
Multiply by -1 and we get -5<-x<3

add the above with -7<y<9

-12<y-x<12... and we are done!!

Hope this helps.

hey killerdrummer, that's awesome!

Thanks Mate aka "whats_in_the_store"!!