If 0

Question

If 0<2x+3y<50 and -50<3x+2y<0 , then which of the following must be true? I. x>0 II. y>0 III. x

dineshril · Accepted Answer

Given 0<2x+3y<50 and -50<3x+2y<0 Since => 0<2x+3y and => 3x+2y<0 Therefore => 3x+2y<2x+3y => OR 3x-2x<3y-2y => OR xboth CANNOT be +ve since given 3x+2y<0 and =>both CANNOT be -ve since given 2x+3y>0 Therefore =>both are OPPOSITE sign =>AND Since x0 so Satisfy II Option E Regards Dinesh

niks18 · Answer

0<2x+3y<50 => 2x+3y is positive ---------(1) -50<3x+2y<0 =>3x+2y is negative -----------(2) notice that by adding x-y to equation (1) it becomes equation (2) i.e a negative value. so x-y<0 => xx , so if y is negative then x has to be negative which will mean that 2x+3y<0 which is not possible. So we can say that y must be positive. Statement II must be true. Option E

MathRevolution · Answer

=>

When we add the two inequalities  0<2x+3y<50  and  -50<3x+2y<0 , we obtain  -50<5x+5y<50 , or  -20<-2x-2y< 20 .

Statement I. 
Adding the two inequalities  -50<3x+2y<0  and  -20<-2x-2y< 20  yields  -70<x<20 . 
So x may not be greater than zero.
Statement I may not be true.

Statement II.
By multiplying all sides of  0<2x+3y<50  by  -3 , we have  -150<-6x-9y< 0 .
By multiplying all sides of  -50<3x+2y<0  by  2 , we have  -100<6x+4y< 0 .
By adding the above inequalities, we have  -250<-5y<0  or  0<y<50 .

Statement II is true.

Statement III.
Since  0<2x+3y<50  is equivalent to  -50<-2x-3y<0  and  -50<3x+2y<0 , adding the two inequalities yields
  -100<x-y<0 . This implies that  x < y .
Statement III must be true.

Therefore, the answer is E.

Answer : E

niks18 · Answer

Hi  MathRevolution ,

Need a clarity in Statement II.

if x & y both can be negative then how will they both satisfy  0<2x+3y<50  and  -50<3x+2y<0  simultaneously? Negative x & negative y will not satisfy the  0<2x+3y<50

MathRevolution · Answer

Yes, you are right.
The solution is fixed. Please look at the above solution again.

finisher009 · Answer

Steps to solve:

1. Subtract the second inequality from the first. A fast way to see the relationship between x and y is to subtract the expressions.
• `(2x + 3y)` is a positive number.
• `(3x + 2y)` is a negative number.
• So, `(2x + 3y) - (3x + 2y)` must be `(Positive) - (Negative) = Positive`.
• `(2x + 3y) - (3x + 2y) = y - x`.
• Since `y - x` must be positive, `y > x` or `x < y`.
• Statement III (x < y) is TRUE. This eliminates options A and B.

2. Eliminate x to find the sign of y (or vice-versa). To determine the signs of x and y, create opposite coefficients for one variable.
• Multiply the first inequality by 3: `6x + 9y` is positive.
• Multiply the second inequality by -2 (this flips the sign): `-2(3x + 2y)` becomes positive. So, `-6x - 4y` is positive.
• Now, add these two new positive expressions: `(6x + 9y) + (-6x - 4y) = 5y`.
• Since `(Positive) + (Positive) = Positive`, `5y` must be positive. Therefore, `y > 0`.
• Statement II (y > 0) is TRUE.

3. Check Statement I. Since we know `y > 0` and `x < y`, let’s look at the second inequality: `3x + 2y < 0`.
• We can rewrite this as `3x < -2y`.
• Since `y` is positive, `-2y` is a negative number.
• If `3x` is less than a negative number, `x` must be negative.
• Statement I (x > 0) is FALSE.

The statements that must be true are II and III. The correct answer is E.

If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr

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