If q, s, and t are all different numbers, is q < s < t ?

(1) t - q = |t - s| + |s - q|

(2) t > q

If q, s, and t are all different numbers, is q < s < t ?

(1) t - q = |t - s| + |s - q|

Notice that the right hand side is positive (it's the sum of two absolute values, so two non-negative values, in fact, in our case two positive values, since we know that the variables are distinct). Thus the left hand side must also be positive, which means that t > q. So, we can have 3 cases for s:

a. ---s---q-------t-------
In this case \(s < q < t\):
\(t - s > 0\) and \(s - q < 0\), which would mean that \(|t - s| = t -s\) and \(|s - q| = -(s - q)\) (recall that |x| = x when x > 0 and x = -x when x <= 0).
So, \(|t - s| + |s - q| = (t -s) - (s - q) = t - 2s + q\).

So, in his case we'd have \(t - q = t - 2s + q\) or \(q=s\). But we are told that q, s, and t are all different numbers, so this case is out.

b. -------q---s---t-------
In this case \(q < s < t\):
\(t - s > 0\) and \(s - q > 0\), which would mean that \(|t - s| = t -s\) and \(|s - q| = s - q\). So, \(|t - s| + |s - q| = (t -s) + (s - q) = t - q\).

This matches the info given in the statement.

c. -------q-------t---s---
In this case \(q < t < s\):
\(t - s < 0\) and \(s - q > 0\), which would mean that \(|t - s| = -(t -s)\) and \(|s - q| = s - q\). So, \(|t - s| + |s - q| = -(t -s) + (s - q) = -t + 2s - q\).

So, in his case we'd have \(t - q = -t + 2s - q\) or \(t=s\). But we are told that q, s, and t are all different numbers, so this case is out.

Only q < s < t case is possible. Sufficient.

(2) t > q. Not sufficient.

Answer: A.
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We want to answer whether q < s < t and (2) says that t > q, NOT q > t.
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We need to determine whether q < s < t, given that q, s, and t are all different numbers.

Statement One Alone:

t - q = |t - s| + |s - q|

Since q, s, and t are different numbers, both |t - s| and |s - q| are positive quantities, and their sum |t - s| + |s - q| will also be positive. This also makes the left-hand side t - q positive. Since t - q > 0, we have t > q.

We know t > q, but we still have to determine whether s is between them. That is, is q < s < t? We have three scenarios to consider.

(1) If q < s < t, then t > s and s > q, and then:

t - q = t - s + s - q

t - q = t - q

We see that this equation holds true: t - q = |t - s| + |s - q|, and furthermore q < s < t.

(2) If s < q < t, then t > s and q > s, and thus t -s is positive while s - q is negative, and we have:

|t - s| + |s - q|

t - s + [-(s - q)]

t - s - s + q

t - 2s + q ≠ t - q

Since t - 2s + q ≠ t - q, the equation does not hold and we can’t have s < q < t.

(3) If q < t < s, then s > t and s > q, and thus t - s is negative while s - q is positive, and we have:

|t - s| + |s - q|

-(t - s) + s - q

-t + s + s - q

-t + 2s - q

Since -t + 2s - q ≠ t - q, we see that the equation does not hold, so we can’t have q < t < s.

We see that only scenario 1 is true if t - q = |t - s| + |s - q,| and we do have q < s < t. Statement one alone is sufficient.

Statement Two Alone:

t > q

We know t > q, but we still have to determine whether s is between them. It’s possible that q < s < t, but it is also possible that s < q < t or q < t < s. Since we don’t know anything about s, we can’t determine which case is valid. Statement two alone is not sufficient.

Answer: A
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Bunnuel I tried it in this way . Is it right or wrong

given If q, s, and t are all different numbers, is q < s < t ?
or is 0<s-q<t-q?
Since i am doing the same operation on all the parts on inequality it doesn't violates any rule i suppose

ST 1 t - q = |t - s| + |s - q|
Here t-q is positive ( sum of two mod's) and also greater than s-q as something is been added to |s - q|

Now this equation can be solved in two ways only either same sign or opposite sign of mods
same sign leads t-q= t-s +s-q lhs=rhs therefore 0<s-q<t-q

different signs lead to t-q= t-s -s+q ,,,,,, not possible

or t-q = -t+s+s-q ,,,,,, again lhs is not equal to rhs . this means that the case s-q<0<t-q doesnt holds

Hence st 1 is sufficient
st 2 clearly not suficient hence a

Would anyone please explain this question by plugging in value ?

Thanks in advance.
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Bunuel can you pls explain how from this \(t - q = t - 2s + q\) you deduce that \(q=s\)

You got the answer right, but your approach wouldn't qualify statement 1. Am I wrong to think that way?

1.
one way:
q= 1, s= 2, and t= 3

t-q = lt - sl + ls - ql

3-1 = l3 - 2l + l2 - 1l
2 = l1l + l1l sufficient

second way:
q= 1, s = 3, and t = 2

t-q = lt - sl + ls - ql

2-1 = l2-3l + l3-1l

1 = l1l + l2l

1 does not equal 3, therefore choosing q=1, s=3, and t=2 cannot be right numbers to verify. I get it that t has to be great than s since second way doesn't yield the equation right. But how do we ensure that this will always hold true for any numbers we use?

Given: q, s, and t are all different numbers

Target question: Is q < s < t ?

Statement 1: t - q = |t - s| + |s - q|
Since q, s, and t are all different numbers, we know that |t - s| is POSITIVE, and |s - q| is POSITIVE.
So, t - q = some positive number
From this we can conclude that: q < t
On the number line we have something like this:

From here we need only determine whether s is between q and t

To help us we can use a nice property that says: |x - y| = the distance between x and y on the number line
For example: |3 - 10| = 7, so the distance between 3 and 10 on the number line is 7

So, the statement "t - q = |t - s| + |s - q|" tells us that: (the distance between t and q) = (the distance between t and s) + (the distance between s and q)
The ONLY time this equation holds true is when is between q and t

Given this, it MUST be the case that q < s < t

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: t > q
Since there is no information about s, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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Hello ccooley, can you let me know that for statement 1, why you did not consider a case where s was greater than both q and t?