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# If x>0, and two sides of a certain triangle

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If x>0, and two sides of a certain triangle [#permalink]  30 Aug 2016, 02:20
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29% (01:38) correct 70% (01:16) wrong based on 75 sessions
If x>0, and two sides of a certain triangle have lengths 2x+1 and 3x+4 respectively, which of the following could be the length of the third side of the triangle?

Indicate all possible lengths.

A) 4x+5
B) x+2
C) 6x+1
D) 5x+6
E) 2x+17
[Reveal] Spoiler: OA
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Re: If x>0, and two sides of a certain triangle [#permalink]  30 Aug 2016, 04:35
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Expert's post
Hi,

I think B should be one of the solutions too

Let the triangle have side named A, B and C

A= 2x+1 and B= 3x+4

A+B= 5x+5

Now we know from triangle inequality

C < A +B

Option A

4x+5 < 5x+5

=>x > 0

Option B

x +2 < 5x+5

=> -4x < 3 => x > $$\frac{-3}{4}$$.

There is also possible as long as x >0 it is also x > $$\frac{-3}{4}$$

Option C

6x+1 < 5x+5

x < 4

Also a possible solution exits such that 0 < x < 4.

Option D is not possible.

Option E is also similarly possible.
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Re: If x>0, and two sides of a certain triangle [#permalink]  30 Aug 2016, 05:10
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Let the third side be y
then (3x + 4) – (2x + 1) < y < (3x + 4) + (2x + 1)
=> x + 3 < y < 5x + 5

As with option B, x+2 is always less than x+3, therefore B is not possible.

And with option E, if we choose x=5, then 2x+17= 27
=> from inequality, 8<27<30
Hope, the solution is clear!
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Re: If x>0, and two sides of a certain triangle [#permalink]  11 Nov 2018, 08:19
Hey I got the answer as A, C ,E..I took x as 1,3 and 8... But my workbook still shows the answer as incorrect. Can somebody help me with this question.
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Re: If x>0, and two sides of a certain triangle [#permalink]  11 Nov 2018, 09:24
Expert's post
Reetika1990 wrote:
Hey I got the answer as A, C ,E..I took x as 1,3 and 8... But my workbook still shows the answer as incorrect. Can somebody help me with this question.

The answer is correct as A, C and E.
the equations can be formed in following way..
(I) the third side is less than the sum of the other two sides...
so third side < (2x+1) + (3x+4) or third side < 5x+5
(II) the third side is greater than the difference of the other two sides...
so third side > |(2x+1) - (3x+4)| or > |x+3|
Therefore, equation becomes $$x+3 < third side < 5x+5$$
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: If x>0, and two sides of a certain triangle [#permalink]  12 Sep 2019, 11:49
Hi All ,
While 4x+5 , 6x+1 , and 2x +17 are mentioned as correct answers for x>0 there might e challenge

X tending to 0 e.g. 0.01
2x+1 = 1.02
3x+4 = 4.03

here 6x+1 as third side will become 1.06 thus three sides as 1.02,1.06, 4.03 which is not possible
here 2x+17 = 17.02 which again is not possible

Again if x is big as x=1000
2x+1 =2001
3x+4 =3004
6x+1 becomes 6001

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Re: If x>0, and two sides of a certain triangle [#permalink]  15 Oct 2019, 06:57
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x+3<3rd side length<5x+5
let's say x=1000
1003<3rd side<5005
option E: 2(1000)+17= 2017 which falls right into the region. N.B if you consider small values for x, ie 2/3 then option E is invalid. but the ques asks for which of the following could be** the length. not which of the following must be**. you have to go for every possible way out there.
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Re: If x>0, and two sides of a certain triangle [#permalink]  04 Jun 2020, 18:31
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phoenixio wrote:
If x>0, and two sides of a certain triangle have lengths 2x+1 and 3x+4 respectively, which of the following could be the length of the third side of the triangle?

Indicate all possible lengths.

A) 4x+5
B) x+2
C) 6x+1
D) 5x+6
E) 2x+17

solution:
of course 3x+4 > 2x+1 so, third side of the triangle is 3X+4-(2x+1)<third_side<3x+4+2x+1.
which gives x+3<third_side<5x+5.
note: question asked which "could be" the length of third side.
x>0:
A) x+3<4x+5<5x+5 which is true.

B) x+3<x+2<5x+5 never true.

C) x+3<6X+1<5x+5 which is not true for all value of x, but is true for x<4. so, could be true.

D) x+3<5x+6<5x+5 never true.

E) x+3<2x+17<5x+5 which is not true for all value of x, but is true for x>=5.
A,C,E
Re: If x>0, and two sides of a certain triangle   [#permalink] 04 Jun 2020, 18:31
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