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# If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true?

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If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
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Bunuel wrote:
If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true?

I. x > y
II. x < y
III. x > 3

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Solution:

We have $$3x + 5y < 15.....(i)$$
and $$5x + 3y > 15$$ or $$-5x-3y<-15......(ii)$$ (We do this because we need to keep the inequality sign same to do the addition)

When we see the options, we see that we need to compare x and y and the value of x if it is greater than 3 or not

Adding inequality $$i$$ and $$ii$$, we get:

$$3x-5x+5y-3y<15-15$$
$$⇒-2x+2y<0$$
$$⇒2y<2x$$
$$⇒y<x$$

This means statement I i.e., $$x>y$$ MUST BE true. Thus we can eliminate options B, C and D

Now we have $$3x + 5y < 15.....(i)$$ and $$-5x-3y<-15......(ii)$$

Multiplying equation (i) with 3, equation (ii) with 5, and adding the equations, we get:

$$9x+15y-25x-15y<45-75$$
$$⇒-16x<-30$$
$$⇒16x>30$$
$$⇒x>\frac{30}{16}$$
$$⇒x>1.\times \times$$

So, statement III i.e., $$x>3$$ is NOT a MUST. It can be true but not a MUST.

Hence the right answer is Option A

Originally posted by SaquibHGMATWhiz on 06 Sep 2022, 23:06.
Last edited by SaquibHGMATWhiz on 08 Sep 2022, 03:18, edited 2 times in total.
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Re: If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
Thank you so much for writing the detailed solution.
The values we got is x > 1.8
y < x
when we plug these values in the inequalities :
assume x=2
y=1
5(x)+3(y)>15
5(2)+3(1)>15 which is not true. Why is this happening...
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If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
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AmritaISBAspirant wrote:
Thank you so much for writing the detailed solution.
The values we got is x > 1.8
y < x
when we plug these values in the inequalities :
assume x=2
y=1
5(x)+3(y)>15
5(2)+3(1)>15 which is not true. Why is this happening...

This is a very common yet interesting mistake that a lot of test takers make. In the case of inequalities, plugging values doesn't really work in the same way as it does for linear equations.

Let's understand why.

Firstly, let us try and get the range of y in the same way we got the range of x
We have $$3x + 5y < 15.....(i)$$ and $$-5x-3y<-15......(ii)$$
Multiplying equation (i) with 5, equation (ii) with 3, and adding the equations, we get:
$$15x+25y-15x-9y<75-45$$
$$⇒16y<30$$
$$⇒y<\frac{30}{16}$$
$$⇒y<1.8$$

Now does it mean we can take any values of x > 30/18 and any value of y < 30/18 and the inequalities will be satisfied? No.

Let us also understand this using a graph which should give you a lot clearer idea.

Attachment:

graphinequality.png [ 115.13 KiB | Viewed 2267 times ]

The red part is the region of 3x + 5y < 15 while the blue is the region of 5x + 3y > 15 and the overlapping part is what we are concerned with.

If you look closely, coordinates like (2, 1) do not lie in the overlapping range.

The idea behind is really simple.

All the values in the overlapping range will follow the trend of x > y, x > 1.8, and y < 1.8.
However, all the values following the trend of x > y, x > 1.8, and y < 1.8 will not necessarily fall in the overlapping range.

It is similar to "all alkalies are bases, but all bases are not alkali"

Hope this helps.

Originally posted by SaquibHGMATWhiz on 07 Sep 2022, 22:35.
Last edited by SaquibHGMATWhiz on 08 Sep 2022, 03:21, edited 1 time in total.
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Re: If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
I can't thank you enough for this explanation!!!!
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Re: If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
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Re: If 3x + 5y < 15 and 5x + 3y > 15, which of the following must be true? [#permalink]
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