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If \(-3 \lt x \lt 5\) and \(-7 \lt y \lt 9\), which of the following represents the range of all possible values of \(y-x\)?

A. \(-4 \lt y-x \lt 4\) B. \(-2 \lt y-x \lt 4\) C. \(-12 \lt y-x \lt 4\) D. \(-12 \lt y-x \lt 12\) E. \(4 \lt y-x \lt 12\)

To get max value of \(y-x\) take max value of \(y\) and min value of \(x\): \(9-(-3)=12\);

To get min value of \(y-x\) take min value of \(y\) and max value of \(x\): \(-7-(5)=-12\);

Hence, the range of all possible values of \(y-x\) is \(-12 \lt y-x \lt 12\).

Answer: D

This question is good but it would be interesting to see a question with the same guidelines as to find the range of values but with the variables x/y or both with inclusive numbers like -3<=x<5 and -7<y<=9.

Also if possible, kindly suggest similar problems.

If \(-3 \lt x \lt 5\) and \(-7 \lt y \lt 9\), which of the following represents the range of all possible values of \(y-x\)?

A. \(-4 \lt y-x \lt 4\) B. \(-2 \lt y-x \lt 4\) C. \(-12 \lt y-x \lt 4\) D. \(-12 \lt y-x \lt 12\) E. \(4 \lt y-x \lt 12\)

To get max value of \(y-x\) take max value of \(y\) and min value of \(x\): \(9-(-3)=12\);

To get min value of \(y-x\) take min value of \(y\) and max value of \(x\): \(-7-(5)=-12\);

Hence, the range of all possible values of \(y-x\) is \(-12 \lt y-x \lt 12\).

Answer: D

This question is good but it would be interesting to see a question with the same guidelines as to find the range of values but with the variables x/y or both with inclusive numbers like -3<=x<5 and -7<y<=9.

Also if possible, kindly suggest similar problems.

another approach: sum up the 2 inequalities. but mind that the inequalities for such operaton should be looking the same way.

since we need to find y-x, we need to figure out how to transform the x-inequality into the required form, i.e. -x: 1. multiply by (-1): we get +3>-x>-5 (remember to flip direction when * -1) 2. still is not looking the same way, therefore: -5<-x<3 is just the same expression of the number line fragment

The questions states that the values are less than or greater than. but the answer explanation uses values that are greater than or EQUAL TO!! The explanation uses values that are not in the range of values given in the question

The questions states that the values are less than or greater than. but the answer explanation uses values that are greater than or EQUAL TO!! The explanation uses values that are not in the range of values given in the question

The questions states that the values are less than or greater than. but the answer explanation uses values that are greater than or EQUAL TO!! The explanation uses values that are not in the range of values given in the question

The OE is correct.

Consider the following approach, we have -3<x<5 and -7<y<9,

Add y<9 and -3<x --> y-3<9+x --> y-x<12; Add -7<y and x<5 --> -7+x<y+5 --> -12<y-x;

I'm not doubting the question itself, but when other people raised issues about the explanation, i think it's reasonable to doubt why the maximum VALUE of y is 9, and minimum VALUE of x is -3. Your wording is totally misguiding. You should say the maximum RANGE of y, rather than the value. The maximum value in this case is not 9, it's 8.99. Since the question is saying that the range is <>, not <=,>=.

I'm not doubting the question itself, but when other people raised issues about the explanation, i think it's reasonable to doubt why the maximum VALUE of y is 9, and minimum VALUE of x is -3. Your wording is totally misguiding. You should say the maximum RANGE of y, rather than the value. The maximum value in this case is not 9, it's 8.99. Since the question is saying that the range is <>, not <=,>=.

The question and the solution are absolutely correct. I think your doubt is addressed here: m26-184451.html#p1655617

I think a much simpler way to solve this problem is to simply subtract two inequlities, according to the inequalities subtraction rules: (-7<y<9)-(5>x>-3) <----last inequality simply reversed. And we get the same answer -12<y-x<12

I made a mistake and subtracted (1) from (2) (since signs are similar in two statements) giving -4 < (y-x) < 4

I agree with the OE but can someone explain if the above allowed and if yes what information does it convey?

No, it is wrong and you can simply test it with numbers..

(1) -3<x<5 ......let x be the lowest possible -2.99999 (2) -7<y<9 ...... let y be the MAX, so 8.99999 y-x = 8.99999-(-2.99999) = 10 and 10 is not in the range -4<y-x<4

you can ofcourse ADD the equations..

(1) -3<x<5 (2) -7<y<9 so -3+(-7)<x+y<5+9......-10<x+y<14

so to sum it all 1) if you are adding, take highest values and add them and for lower range add lowest values and add them 2) If you are subtracting, take highest of one and lowest of other
_________________

I made a mistake and subtracted (1) from (2) (since signs are similar in two statements) giving -4 < (y-x) < 4

I agree with the OE but can someone explain if the above allowed and if yes what information does it convey?

No, it is wrong and you can simply test it with numbers..

(1) -3<x<5 ......let x be the lowest possible -2.99999 (2) -7<y<9 ...... let y be the MAX, so 8.99999 y-x = 8.99999-(-2.99999) = 10 and 10 is not in the range -4<y-x<4

you can ofcourse ADD the equations..

(1) -3<x<5 (2) -7<y<9 so -3+(-7)<x+y<5+9......-10<x+y<14

so to sum it all 1) if you are adding, take highest values and add them and for lower range add lowest values and add them 2) If you are subtracting, take highest of one and lowest of other