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Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.

\[\large (X-1)(X-3)(X-5)(X-7)\ldots (X-97) < 0 \]

How many positive integers \(X\) satisfy the inequality above?

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\[\left| 1-\left| 1-\left| x-1 \right| \right| \right| =\alpha \]

Find the value of \(\alpha\) for which this equation has exactly 4 distinct solutions. If the value of \(\alpha\) is \(A\), then find \(2A\).

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Find the area of the region bound by the equation \[|x+2y|+|2x-y|=10\]

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Given the graph \( y = \ln x \), which of these statements describes the transformations to get the graph of \( y = \ln (4x^2 + 4x + 1) \) for \( x > - \frac{1}{2} \)?

\(\quad \text{(1)}\) Translate to the left by 1 and up by \( \ln 4 \), then scale vertically by 2.

\(\quad \text{(2)}\) Translate to the left by \(\frac{1}{2}\) and up by \( \ln 2 \), then scale vertically by 2.

\(\quad \text{(3)}\) Translate to the left by 1 and up by \( \ln 2 \), then scale vertically by 2.

\(\quad \text{(4)}\) Translate to the left by \( \frac{1}{2} \) and up by \( \ln 4 \), then scale vertically by 2.

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