Find all possible values of X | Solving Inequalities | Important math skills explained

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Find all possible values of X | Solving Inequalities | Important math skills explained

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[marcfrantz]

Thanks for this nice problem, and thanks to Chris Farmer for the comment below. Like Chris, I was disturbed by the possibility of x approaching 4 from above, which leads to a contradiction. I believe the estimate can be sharpened and the contradiction avoided by simply relabeling the diagram. Specifically, relabel the sides with single tickmarks as 1, and label the sides with double tickmarks as (say) k, for any k>0. Applying the triangle inequality to the green triangles, we have x+2>1+k and 2x-8>1+k. Now 2x-8 is the shorter side, so both are true if 2x-8>1+k. Since k can be arbitrarily close to zero, we can can only be sure that 2x-8>1. Solving this for x gives x>9/2=4.5. Coupling this with the original result x<10, we get the allowable range of values for x as 4.5<x<10. I believe this is as sharp an estimate as we can get, and it eliminates the possibility of x approaching 4 from above.

[HappyFamilyOnline]

Great explanation👍
Thanks for sharing😊

[TheLango]

I do not understand why you are using the Angle Side Relation Theorem at all. My understanding of this theorem is that, in any given triangle, the longest side will be opposite the largest angle and the shortest side will be opposite the smallest angle. But, you are evaluating two different triangles. You are basically looking for "boundary conditions" of the same relative angle within two different triangles. Forgive me if I am not using the proper terminology (it's been over 30 years since my last math class in school). What I mean is, the angle you are evaluating is between the two legs whose lengths are the same in both triangles. I really don't understand how the ASR Theorem is relevant to this example in any way. I must be missing an important point here. Thank you!

[chrisfarmer]

If this is the possible range of X, how does that hold up near the extreme of X approaching 4? As X approaches 4 from the positive infinity side the side of (2X-8) approaches 0 so the other sides of the triangle would also be approaching 0. But on the other green triangle as X approaches 4 X+2=6 so the single slash and double slash lines cannot be approaching 0. Lunch break is over so I'll have to think about this more but it seems like there is only 1 possible solution intuitively.

[Xrayvis]

Why do you assume 88° is the smaller angle?

[Okkk]

The problem should be stated “find all possible values of x”.