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# In the triangle above, if 80 ≤ y ≤ 100, which of the following must be

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Math Expert
Joined: 02 Sep 2009
Posts: 46284
In the triangle above, if 80 ≤ y ≤ 100, which of the following must be [#permalink]

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05 Sep 2017, 23:52
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In the triangle above, if 80 ≤ y ≤ 100, which of the following must be true?

(A) 30 ≤ x ≤ 40
(B) 40 ≤ x ≤ 50
(C) 50 ≤ x ≤ 60
(D) 60 ≤ x ≤ 70
(E) 70 ≤ x ≤ 80

Attachment:

2017-09-06_1035_001.png [ 11.09 KiB | Viewed 875 times ]

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Re: In the triangle above, if 80 ≤ y ≤ 100, which of the following must be [#permalink]

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06 Sep 2017, 00:40
By figure y+x+x=180
y+2x=180

Given: 80 ≤ y ≤ 100

Min value of x
=> y=100 => 100+2x=180 => x=80/2 =40

Max value of x
=> y=80 => 80+2x=180 => x=100/2 =50

ie 40 ≤ x ≤ 50

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Re: In the triangle above, if 80 ≤ y ≤ 100, which of the following must be [#permalink]

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06 Sep 2017, 03:55
Bunuel wrote:

In the triangle above, if 80 ≤ y ≤ 100, which of the following must be true?

(A) 30 ≤ x ≤ 40
(B) 40 ≤ x ≤ 50
(C) 50 ≤ x ≤ 60
(D) 60 ≤ x ≤ 70
(E) 70 ≤ x ≤ 80

Attachment:
2017-09-06_1035_001.png

Sum of angles of triangle = 180 deg
So, 2x+y = 180 -> 2x = 180 - y ...........(i)
Also 80 ≤ y ≤ 100.............(ii)

From (i) & (ii), 80 ≤ 2x ≤ 100
40 ≤ x ≤ 50

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Re: In the triangle above, if 80 ≤ y ≤ 100, which of the following must be [#permalink]

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11 Sep 2017, 11:01
Bunuel wrote:

In the triangle above, if 80 ≤ y ≤ 100, which of the following must be true?

(A) 30 ≤ x ≤ 40
(B) 40 ≤ x ≤ 50
(C) 50 ≤ x ≤ 60
(D) 60 ≤ x ≤ 70
(E) 70 ≤ x ≤ 80

Attachment:
2017-09-06_1035_001.png

Since y + 2x = 180:

2x = 180 - y

x = 90 - y/2

We are given that 80 ≤ y ≤ 100. If y = 80, then x is at its maximum, i.e., x is at most 90 - 80/2 = 50. If y = 100, then x is at its minimum, i.e., x is at least 90 - 100/2 = 40. Thus, x is between 40 and 50 inclusive, i.e., 40 ≤ x ≤ 50.

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In the triangle above, if 80 ≤ y ≤ 100, which of the following must be [#permalink]

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12 Sep 2017, 08:32
Bunuel wrote:

In the triangle above, if 80 ≤ y ≤ 100, which of the following must be true?

(A) 30 ≤ x ≤ 40
(B) 40 ≤ x ≤ 50
(C) 50 ≤ x ≤ 60
(D) 60 ≤ x ≤ 70
(E) 70 ≤ x ≤ 80

Attachment:
2017-09-06_1035_001.png

If 80 ≤ y ≤ 100, which of the following must be true?

1. Answer choices all contain x, so find y in terms of x

y + 2x = 180
y = 180 - 2x

2. Substitute (180 - 2x) for y in the inequality

80 ≤ 180 - 2x ≤ 100

3. Solve for x (subtract 180 and divide by -2, switching direction of inequality sign)

-100 ≤ -2x ≤ -80

50 ≥ x ≥ 40

40 ≤ x ≤ 50

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In the triangle above, if 80 ≤ y ≤ 100, which of the following must be   [#permalink] 12 Sep 2017, 08:32
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# In the triangle above, if 80 ≤ y ≤ 100, which of the following must be

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