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Re: If x>0, and two sides of a certain triangle [#permalink]
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]
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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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Re: If x>0, and two sides of a certain triangle [#permalink]
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

Not so sure about the answers now ?
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Re: If x>0, and two sides of a certain triangle [#permalink]
3
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]
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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 :clap: :clap:
:twisted: :twisted:
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Re: If x>0, and two sides of a certain triangle [#permalink]
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C and E should be out of the answers because we don't know the real value of X
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Re: If x>0, and two sides of a certain triangle [#permalink]
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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


IMPORTANT PROPERTY: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between A and B < length of third side < SUM of A and B

ASIDE: If x > 0, then we can be certain that 3x+4 is greater than 2x+1

Let's plug the two given lengths into the above property to get: (3x+4) - (2x+1) < length of 3rd side < (3x+4) + (2x+1)
Simplify to get: x + 3 < length of 3rd side < 5x + 5

So any length that could satisfy the above inequality will be a possible length.
So let's start testing!!!

A) 4x+5
Plug this value into our inequality: x+3 < 4x+5 < 5x+5
If x = 1, the inequality holds.
So answer choice A COULD be the length of the third side.

B) x+2
Plug this value into our inequality: x+3 < x+2 < 5x+5
Let's focus on this part of the inequality: x+3 < x+2
Subtract x from both sides to get: 3 < 2
At this point, we can conclude that there are no possible values of x that will satisfy the inequality.
So answer choice B CANNOT be the length of the third side.

C) 6x+1
Plug this value into our inequality: x+3 < 6x+1 < 5x+5
If x = 1, the inequality holds.
So answer choice C COULD be the length of the third side.

D) 5x+6
Plug this value into our inequality: x+3 < 5x+6 < 5x+5
Let's focus on this part of the inequality: 5x+6 < 5x+5
Subtract 5x from both sides to get: 6 < 5
At this point, we can conclude that there are no possible values of x that will satisfy the inequality.
So answer choice D CANNOT be the length of the third side.

E) 2x+17
Plug this value into our inequality: x+3 < 2x+17 < 5x+5
Don't forget that all we need is one value of x that satisfies the above any quality in order to show that the answer choice COULD be the length of the third side.
After a bit of testing I found that when x = 10, the inequality holds.
So answer choice E COULD be the length of the third side.

Answer: A, C, E
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Re: If x>0, and two sides of a certain triangle [#permalink]
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