Graphics Coordinate: Is This Correct?

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The main topic of the discussion is the analysis of the inequality \((x - 1)^4 < (x - 1)\) and its graphical interpretation. The conclusion reached is that the inequality holds true for the interval \(x \in (1, 2)\). The graphs of the functions \(y = (x - 1)^4\) and \(y = x - 1\) intersect at \(x = 1\) and \(x = 2\), confirming that \(y = (x - 1)^4\) lies below \(y = x - 1\) within this interval. The factorization of the equation \((x - 1)^4 - (x - 1) = 0\) is essential for determining the points of intersection.

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View attachment 2983

The answer is E.

Since the line is passing the parable at x = 1 and 2 I used between these values to satisfy the inequality(x − 1)4< (x − 1)X = 1,5

(1,5 -1)4 < (1,5 -1)
0,0625 < 0,5

Is this correct?
 

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I would begin by finding the $x$-values for which the two graphs intersect. We can do this by equating the two functions:

$$(x-1)^4=x-1$$

$$(x-1)^4-(x-1)=0$$

Now, what do you get when you factor?
 
I was going to type exactly what MarkFL did, but he beat me at it :p

I also want to note that the question has graciously graphed the curves for us, and by inspection, we see that $y=(x-1)^4$ is under the line $y=x-1$ on the interval $(1, 2)$ only, which is the only interval that satisfies your inequality.
 

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