How do I convince students in high school for which

Helen Lewis 2022-01-04 Answered
How do I convince students in high school for which this equation: \(\displaystyle{2}^{{x}}={4}{x}\) have only one solution in integers that is \(\displaystyle{x}={4}\)?

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encolatgehu
Answered 2022-01-05 Author has 1417 answers
Hint:
plot the graph of \(\displaystyle{y}={2}^{{x}}\) and \(\displaystyle{y}={4}{x}\) and shows that the only other solution is between 0 and 1.
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Jeffery Autrey
Answered 2022-01-06 Author has 3319 answers
If your intention is to 'convince' and not to prove, I'd draw a graph with the functions \(\displaystyle{y}={2}^{{x}}\ \text{ and }\ {y}={4}{x}\).The growth rate of each function should make clear that they intersect only at two points, being the first between 0 and 1 (and hence, not being an integer).
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Vasquez
Answered 2022-01-11 Author has 8850 answers

For positive n, we have two growing sequences
1,2,4,8,16,32,64,128,256⋯
0,4,8,12,16,20,24,28,32⋯
This shows them that the ''curves'' cross each other at 16, and it seems that the first grows faster.
Indeed, taking the ratios of successive terms
\(\frac{2^{n+1}}{2^n}=2>\frac{4(n+1)}{4n}=1+\frac{1}{n}\)

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The following data was collected about students in Mr. Rexinger's high school statistics class. Wearing jeansNot wearing jeans Has long hair(below chin)77 Does not have long hair 513

a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain.

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c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events \(\{\text{not having long hair}\}\) and \(\{\text{wearing jeans}\}\) associated for the students in Mr. Rexinger's class today? Explain the independence relationship using \(P(A\ \text{given}\ B) = P(A)\).

d. Are the events \(\{\text{not having long hair}\}\) and \(\{\text{wearing jeans}\}\) mutually exclusive? Explain.

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