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

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Which of the following inequalities indicates the set of all values of  [#permalink]

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New post Updated on: 16 Jan 2016, 13:07
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which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.

Originally posted by alice7 on 16 Jan 2016, 12:35.
Last edited by ENGRTOMBA2018 on 16 Jan 2016, 13:07, edited 2 times in total.
Reformatted the question and updated the topic
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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 18 Jan 2016, 10:23
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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.

Attachment:
Triangle sides.gif
Triangle sides.gif [ 106.02 KiB | Viewed 4509 times ]


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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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 16 Jan 2016, 13:06
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alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.



Follow posting guidelines (link in my signatures).


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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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 17 Jan 2016, 21:32
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.

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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 17 Jan 2016, 21:37
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alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 17 Jan 2016, 23:27
alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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)
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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 12 Dec 2017, 02:46
VeritasPrepKarishma wrote:
alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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
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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 12 Dec 2017, 03:15
alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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

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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 12 Dec 2017, 04:37
shivamtibrewala wrote:
VeritasPrepKarishma wrote:
alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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


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 ..."
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Re: Which of the following inequalities indicates the set of all values of  [#permalink]

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New post 19 Apr 2018, 14:05
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alice7 wrote:
which of the following inequalities indicates the set of all values of d for which the lengths of the three sides of a triangle can be 3,4, and d?

A) 0<d<1
B) 0<d<5
C) 0<d<7
D) 1<d<5
E) 1<d<7

I end up choosing 3 because 9+16 = 25, so 1,2, and 4 is out.
Now I was debating whether to go for 3 or 5 and couldn't decide.


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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