Which of the following inequalities indicates the set of all values of

It's a lot easier to visualize what's going on than to describe it with words, so I created a little animation to help those who still might be having trouble.



As we can see, the red side can never be shorter than the difference of the other two sides, and never longer than the sum of the other two sides.

So in this problem, the third (red) side can never be less than (4-3)=1, and never more than (4+3)=7.

1 < d < 7

Answer E

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This question is a good way to apply one of the most basic relation between the 3 sides of a triangle. In a triangle (ANY TRIANGLE), any side MUST be greater than the positive difference of the other two sides and less than than the sum of the other 2 sides. Let the sides of a triangle be a,b,c .

Thus,

|a-b| < c < a+b
|b-c| < a < b+c
|c-a| < b < a+c

Thus, if the sides of the triangle are 3,4 and d,

4-3<d<4+3 ---> 1<d<7. Thus E is the correct answer.

Hope this helps.
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IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
difference between sides A and B < third side < sum of sides A and B

So, 4 - 3 < d < 4 + 3
Simplify: 1 < d < 7
Answer: E

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Hi All,

This question is meant to test your understanding of the Triangle Inequality Theorem, but the answer choices are written in a way that you don't actually have to know that math rule to get the correct answer.

We're asked for the range of possible values for the third side of a triangle when the first two sides are 3 and 4...

You're probably all familiar with the 3/4/5 right triangle, so the correct answer MUST include 5 in its range.
Eliminate Answers A, B and D.

For this next part, drawing a picture might help. Draw a line with a length of 4, then draw another line right "on top of" the first line with a length of 3. The length of the first line that is NOT covered by the second line is 4-3 = 1. If the third side was equal to 1, then we would NOT have a triangle - we would have a line right on top of another line. That line with a length of 1 shows us that the third side of the triangle has to be greater than 1.
Eliminate Answer C.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Hi,

D has to be more than 4-3, or 1 otherwise it will become a straight line of 4 length..
D has to be less than 4+3, or 7, otherwise again we will have a straight line of d length..

so 1<d<7
E
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We are already aware of the rule "third side of a triangle must be less than the sum of other two sides"
So if two sides are 3 and 4, the third side must be less than 7.
The same rule also implies that "third side of a triangle must be greater than the difference between the other two sides."

Here is why:
Say the three sides of the triangle are a, b and c.

a + b > c
Third side is greater than sum of other two.
Similarly, b + c > a

c > a - b
Here we see that c, the third side, is greater than the difference between the other two sides.

So if two sides are 3 and 4, the third side must be greater than 1.

Hence we get the range 1 < d < 7.

Answer (E)

Still don't understand the difference between D and E. Any help would be appreciated

The limits for the third side of a triangle is
a. 3rd side of triangle can't be greater than or equal to sum of 2 sides.
b. 3rd side of triangle can't be smaller than or equal to difference of 2 sides.

So, |3-4| <d < 3+4
1 < d< 7

Answer E

Here is the difference between (D) and (E).
(D) doesn't cover the entire range of possibilities. e.g. d can be 6.
3, 4 and 6 form a triangle. The sum of each pair of two sides is greater than the third. (E) covers the entire range.

The question asks for the set of ALL values:
" ... indicates the set of all values of d ..."

Maybe I'm not understanding the logic here correctly. Let's tweak the question and say that we had sides with lenghts 7, 3 and D.

By this formula, (7-3)<D<(7+3) -> 4<D<10.

4 now falls out of range. Can someone explain? Thanks!

Hi Aashay94,

This type of situation is based on the Triangle Inequality Theorem. The simple idea behind this math rule is that when you are forming triangles and have the values of two of the sides, you can determine the 'range' of values for the third side.

The smallest POSSIBLE length for the third side must be greater than the 'positive difference' of the two sides that you have.
The largest POSSIBLE length for the third side must be less than the sum of the two sides that you have.

Thus, with sides of 3, 7 and D....
The smallest possible distance is GREATER than (7 - 3) = 4
The largest possible distances is LESS than (7 + 3) = 10
Thus 4 < D < 10.

The same concept applies to the question at the beginning of this thread (changes the values to 3, 4 and D and you'll see).

GMAT assassins aren't born, they're made,
Rich

Thanks Rich! Appreciate it

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Hi Sir Bunuel

I was able to get the range of the value of d i.e. 1<d<7, But i got confused because i once encountered the similar type of question. As in this question there are two options which include the complete set of probable values of 'd' i.e. Option C and E.

However i marked the option E during my mock.

Question i faced earlier from which i noticed a issue: Actually there was a inequality mentioned and after solving the equation i managed to find the value of x. Let in this case X>5. And one of the option was mentioned X>2, which indeed was the correct answer because X>2 definitely includes the X>5.

How the question i solved earlier is different from this question. Is there any flaw in my thought process. Kindly help me.

If the question were "The three sides of a triangle are 3, 4, and d. Which of the following must be true about d?", then the option "0 < d < 7" would be correct. That's because while for the triangle condition, d needs to be between 1 and 7, any value of d between 1 and 7 also falls within the broader range of 0 to 7.

However, the problem at hand asks a different question. It seeks to pinpoint the exact range for which all possible values of d will form a triangle with sides 3 and 4. In this specific context, "0 < d < 7" is not true because d cannot be less than 1. The precise range capturing all valid d values is "1 < d < 7".
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Theory: Each Side of the Triangle is between the | Difference of the other two sides | and Sum of the other two sides

=> |3 - 4| < d < 3 + 4
=> |-1| < d < 7
=> 1 < d < 7

So, Answer will be E
Hope it helps!
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