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Since A, B and C are three different point, AB, BC,and AC are all positive values. so AC will have a value defines as 0 < AC < 10. because if AC> 10, AB becomes negative. SO this inturn gives the value of AB as : 10 > AB > 0. So AB is less than 10.

Pls tell me where I went wrong!!
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[caption=]Remember: Anything that can go wrong, will go wrong.[/caption]

A , B , and C are points on the plane. Is AB < 10 ?

1. AC + BC = 10 2. AB + AC > 10

Since we already know that individual statements are insufficient, let's combine:

From 1, we know that AC= 10 - BC. Let's substitute this into 2. AB + 10 - BC > 10 =====> AB > BC

From 1, we know that BC < 10, since AC must be some value greater than zero. Combining AB > BC with BC < 10, we have AB > 0, and this is not enough to tell us whether AB < 10.

the answer is E, and a quick dirty way to do it is by plugging in numbers. First thing to note is that it the original question does not say A B or C are distinct, implying that A could equal B and so on. Also, the way the equations are set up it is just easier to think about the points as numbers.

so is stmt 1 enough? AC + BC = 10. plug in A=2, B=0, C=5 in this case AB<10 now plug in A = 5, B =5, C=1 in this case AB>10. Therefore stmnt 1 is not enough.

is stmt 2 enough? AB +AC >10 A = 11, B = 1, C = 1 and AB >10 A = 1, B = 1, C =10 and AB<10 therefore stmnt 2 is not enough

Together? AC + BC = 10 and AB+AC>10 A=3, B = 2, C =2 and AB<10 A=18, B = 2, C =.5 and AB>10

A , B , and C are points on the plane. Is AB < 10 ?

1. AC + BC = 10 2. AB + AC > 10

Since we already know that individual statements are insufficient, let's combine:

From 1, we know that AC= 10 - BC. Let's substitute this into 2. AB + 10 - BC > 10 =====> AB > BC

From 1, we know that BC < 10, since AC must be some value greater than zero. Combining AB > BC with BC < 10, we have AB > 0, and this is not enough to tell us whether AB < 10.

Therefore, E.

Cheers, Der alte Fritz.

the Bunuel approach of course is the faster........but yours is pretty good
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My pick is also E. If we have been provided with some additional information on the location of the points, for example whether it is a triangle, a straight line or something then provided statements could have been sufficient.
_________________

If you found my post useful and/or interesting - you are welcome to give kudos!

But if we solve for the inequality in equation 2, we get AB>10-AC. Since AC cannot be negative or zero, AB will be < 10 no matter if the points are in a triangle or collinear. Isn't the second statement enough and answer B?

If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?

(1) The sum of the lengths of line segments AC and BC is 10 (2) The sum of the lengths of line segments AB and AC is more than 10

Even when we consider both statements together we can not have a definite answer. Consider two examples below:

Attachment:

Number line.png

Answer: E.

This was a tough question for me. Since there are three distinct points, there are 6 ways to arrange those three points. What is an efficient approach to this problem?

It is written that A,B, C are on a plane. So they can be anywhere on the plane. Let's assume they are on a straight line and B is the middle of A and C

A--------B---------C

In that case if AC+BC = 10 then AB is definitely less than 10 (evident from diagram above). So it concludes AB<10 Now let's assume in that scenario C is the middle point between A and B

A--------C---------B

From this image if AC+BC =10 then AB =10 i.e. AB is not <10

So more than 1 answer is possible for 1st option. Hence A not sufficient.

2. AB+AC>10 => AB>10-AC

Let's assume AB = 12 and AC = 2 then upper equation stands true i.e 12>8. In this case AB is not <10 Now assume AB = 3 and AC = 8 then also upper equations stands true i.e. 3>2 But in this case AB<10

So more than 1 answer is possible. Thus not sufficient.

Similarly if we consider both options we will get different results where AB <10 and AB is not less than 10 i.e not sufficient.