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Intern  Joined: 01 Feb 2013
Posts: 9
Location: United States
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If 2s > 8 and 3t < 9  [#permalink]

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5
11 00:00

Difficulty:   25% (medium)

Question Stats: 72% (01:32) correct 28% (01:31) wrong based on 479 sessions

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If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

Originally posted by tulsa on 25 Mar 2013, 12:13.
Last edited by tulsa on 25 Mar 2013, 13:30, edited 2 times in total.
Verbal Forum Moderator B
Joined: 10 Oct 2012
Posts: 583
Re: If 2s > 8 and 3t < 9  [#permalink]

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9
tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

s>4

t<3 (multiply by -1) --> -t>-3

Thus s-t >4-3

or s-t>1. As none of the options given are greater than 1, the answer is none.

A.
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Re: If 2s > 8 and 3t < 9  [#permalink]

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4
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

Solution :
----------
2s > 8 => s > 4
3t < 9 => t < 3

The given values can be represented on the number line as follows :

--t-----------3---------------4--------s-----------
|----+ve---|---------1-----|-+ve-+|

So,s-t will be greater than 1.
Correct option is A.
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Joined: 02 Aug 2009
Posts: 8309
Re: If 2s > 8 and 3t < 9  [#permalink]

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1
Sapient wrote:
Hi,

The Problem description is not precise, could you provide clear problem solving strategy.

Thanks.

If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

we have two equations..
2s > 8 .. s>4, so' s' can be 4.1,5,5.05 etc
and 3t < 9 .. t<3 so 't'can be 2.9999, 2,-1 etc..

if we take the lowest possible difference between s and t, we will take lowest value of s, which is just above 4 and highest value of t, which is just lower to 3..
s-t >4.0000000001 -2.999999999 .... so s-t>1
therefore all values of -1,0,1 are not possible
ans none A
hope it helped
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If 2s > 8 and 3t < 9  [#permalink]

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1
I understand how we get s>4 and t<3, but I'm having a hard time wrapping my head around why t cannot be negative. Can someone please explain?

We don't need to check when t<0 but for the sake of your question, even if t <0 ---> -t>0 and you know that s>0, giving you s-t > 1 for all values of s and t as s>1

Any positive quantity added to a quantity >1 will give you the sum as >1

Consider s= 7, t= -5 or -0.3

Both these cases make it s-t >1.

Hope this helps.
Intern  Joined: 02 May 2013
Posts: 7
Re: If 2s > 8 and 3t < 9  [#permalink]

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Hi,

The Problem description is not precise, could you provide clear problem solving strategy.

Thanks.
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J

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Re: If 2s > 8 and 3t < 9  [#permalink]

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I understand how we get s>4 and t<3, but I'm having a hard time wrapping my head around why t cannot be negative. Can someone please explain?
Intern  B
Joined: 10 Oct 2014
Posts: 19
GPA: 3.47
Re: If 2s > 8 and 3t < 9  [#permalink]

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Thank you. I neglected to utilize the fact that s must be positive when testing cases. Appreciate the help
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Re: If 2s > 8 and 3t < 9  [#permalink]

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Top Contributor
tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

Given: 2s > 8
Divide both sides by 2 to get: s > 4

Given: 3t < 9
Divide both sides by 3 to get: t < 3

NOTE: If we have two inequalities with the inequality symbols facing in the same direction, we can add the inequalities to learn something new.

So, take t < 3 and multiply both sides by -1 to get: -t > -3 [aside: when we divide or multiply both sides of an inequality by a NEGATIVE value, we mist REVERSE the symbol]

We now have:
s > 4
-t > -3

When we ADD these two inequalities, we get:
s - t > 1

If s - t > 1, then:
I) s - t CANNOT equal -1
II) s - t CANNOT equal 0
III) s - t CANNOT equal 1

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Re: If 2s > 8 and 3t < 9  [#permalink]

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2s > 8 and 3t < 9

So, s > 4 and t < 3

Assume
s = 4 + ds
t = 3 - dt

So, s - t = (4 + ds) - (3 - dt)
s - t = 1 + (ds + dt)

So, s - t has to be > 1

None of the given option qualifies.
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Re: If 2s > 8 and 3t < 9  [#permalink]

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tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

We see that s > 4 and that t < 3. Since s is always greater than t, the difference cannot be -1 or zero.

Furthermore, since s > 4 and t < 3, we see that s and t are more than 1 unit apart, so the difference cannot be 1.

Alternate Solution:

Let’s divide each side of 2s > 8 by 2: s > 4

Let’s divide each side of 3t < 9 by -3, paying attention to change the direction of the inequality since we are dividing by a negative number: -t > -3

Let’s add the two inequalities together: s - t > 1

We see that none of the provided numbers is greater than 1.

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Re: If 2s > 8 and 3t < 9  [#permalink]

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mau5 wrote:
tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

s>4

t<3 (multiply by -1) --> -t>-3

Thus s-t >4-3

or s-t>1. As none of the options given are greater than 1, the answer is none.

A.

pushpitkc any idea why do we multiply t<3 by -1 and s>4 leave as it is ? thank you Senior PS Moderator V
Joined: 26 Feb 2016
Posts: 3305
Location: India
GPA: 3.12
If 2s > 8 and 3t < 9  [#permalink]

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1
dave13 wrote:
mau5 wrote:
tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

s>4

t<3 (multiply by -1) --> -t>-3

Thus s-t >4-3

or s-t>1. As none of the options given are greater than 1, the answer is none.

A.

pushpitkc any idea why do we multiply t<3 by -1 and s>4 leave as it is ? thank you dave13 - We have been asked to find the value of s-t. That's the reason for multiplying the inequality involving t with -1.
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Joined: 22 May 2016
Posts: 3747
If 2s > 8 and 3t < 9  [#permalink]

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1
dave13 wrote:
mau5 wrote:
tulsa wrote:
If 2s > 8 and 3t < 9, which of the following could be the value of s-t?

I. -1
II. 0
III. 1

A. None
B. I only
C. II only
D. III only
E. II and III

s>4

t<3 (multiply by -1) --> -t>-3

Thus s-t >4-3

or s-t>1. As none of the options given are greater than 1, the answer is none.
A.

pushpitkc any idea why do we multiply t<3 by -1 and s>4 leave as it is ? thank you Hi dave13 - I hope you have been slaying Quant dragons. At the least, dump some water on their heads. I'm going to expand a little on pushpitkc 's good answer. BTW, I have to add little dots at times to get terms to line up.

We multiply one of the inequalities by -1 because they have signs that point in different directions. The rule: you cannot add inequalities unless their signs point in the SAME direction.

Another rule: multiplying an inequality by any negative number changes the direction of the sign.
Another rule: Multiplying by -1 changes the sign but leaves the numbers and variables the same except with opposite signs.

We are asked to find $$s-t$$. If we multiply $$(t<3)$$by $$-1,$$ we can make $$t$$ negative (hang on) AND flip its sign so it points the same way as that of $$s$$
We've isolated $$s$$ and $$t$$ to get: $$s>4$$ and $$t<3$$

One sign MUST change so we can add. We change the $$t$$ inequality because we will get a sign flip AND a MINUS $$t$$
$$s$$ + $$(-t)$$? Is $$(s-t)$$

Mulitply $$(t<3)$$ by (-1). SIGN flips.
$$(-1*t)>(-1*3)$$
$$-t > -3$$

··$$(s > 4)$$
+$$(-t>-3)$$
------------------
$$s - t> 1$$

Horizontally:
$$s + (-t) = (s-t)$$
$$4 + (-3) = (4-3) = 1$$
The > sign is between, thus $$s - t> 1$$

No answer choices are greater than 1. The answer is A. Hope that helps.

Technically, we can subtract (NOT add) inequalities with different signs. The sign on top controls.
··$$(t<3)$$
-$$(s>4)$$
============
$$t-s<-1$$ ....Oh yay. Now we get to multiply by -1. We need (s-t), not (t-s).
$$(-1*t)-(-1*s)<(-1*-1)$$
$$-t + s>1$$
$$s-t>1.$$ Trust me. ADD. _________________
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If 2s > 8 and 3t < 9  [#permalink]

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visualizing a number line may be helpful.

--------0----1----2----3----4-----

s is somewhere to the right of 4. t is somewhere to the left of 3. We can see the space between them must be greater than 1. All of the choices are not possible. If 2s > 8 and 3t < 9   [#permalink] 22 Mar 2019, 11:44
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