Question 612369: PLEASE help me with this....
one side of a triangle has a length of 5x-2. a longer side has a length of 6x+7. if T represents the length of the third side which of the following is true?
a.x+5 < T <11x+5
b.x+9 < T <11x+5
c.5x-2 < T <6x+7
d.5x+7 < T <6x-2
I had to space out the answers like that because for some reason if i didn't they wouldn't show up correctly.
I've tried triangle inequalities,and i've went to the book and read about triangles to try and solve the LAST question on my homework and i CAN NOT figure this out! I have a picture drawn out of a triangle and labeled sides with the info that was given, im just stuck.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The inequalities involving triangles tell you that the sum of the two shortest sides must be longer than the third side or else a triangle cannot be formed.
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Since you are not given the length of T, the third side, it can vary in length.
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Since T can vary in length, if it is drawn so that it is the longest side, then the two short sides are 5x - 2 and 6x + 7. The sum of these two short sides is:
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(5x - 2) + (6x + 7)
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And when you add them together you get 11x + 5. This sum must be greater than T, so we can write:
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11x + 5 > T
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which is read either as 11x + 5 must be greater than T or T must be less than 11x + 5.
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Let's write it as T must be less than 11x + 5:
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T < 11x + 5
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At this point, notice that only answers (a) and (b) involve 11x + 5 being greater than T. So we can eliminate choices (c) and (d).
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Now let's imagine that we make the length of T smaller and smaller until it is the smallest side in the triangle. This would make the two small sides be T and 5x -2. In order for a triangle to be formed, the sum of these two small sides must be greater than the third side which is now 6x + 7. Let's first add the two small sides:
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T + (5x - 2) which can be written as just T + 5x - 2
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and since this sum must be greater than the third side which is 6x + 7 we can write this inequality as:
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T + 5x - 2 > 6x + 7
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We want to solve this inequality for T, so we need to get rid of the 5x and the minus 2 on the left side of the inequality sign. We can do that by first subtracting 5x from both sides of the inequality sign as shown below:
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T + 5x - 5x - 2 > 6x - 5x + 7
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On the left side the +5x and the -5x cancel each other. And on the right side, the 5x is subtracted from 6x to result in just x. After this, the inequality is reduced to:
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T - 2 > x + 7
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To get rid of the -2 on the left side, we just add 2 to both sides:
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T - 2 + 2 > x + 7 + 2
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On the left side the -2 and +2 cancel each other, and on the right side the +7 and +2 add to +9, making the inequality become:
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T > x + 9
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This is read as T is greater than x + 9 or equivalently as X + 9 is less than T as shown below:
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x + 9 < T
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So we have now developed the two possibilities:
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T < 11x + 5 and x + 9 < T
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And these can be combined and written as
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x + 9 < T < 11x + 5
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This means that answer (b) is the correct choice.
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Hope this helps you to understand the problem and how you can work it to get the answer.
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