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Algebra

Polynomial Inequalities

Polynomial Inequalities: Level 3 Challenges

         

\[ \big(n^2-2\big)\big(n^2-20\big)<0\]

How many integers \(n\) satisfy the inequality above?

Over the ranges

\[ \begin{eqnarray} -1 \leq & v & \leq 1 \\ -2 \leq & u & \leq -0.5 \\ -2 \leq & z & \leq -0.5 \\ \end{eqnarray} \]

what is the range of \(w = \frac {vz}{u} \)?

If \(x\) is an integer that satisfies the inequality

\[ 9 < x ^2 < 99, \]

find the difference between the maximum and minimum possible values of \(x.\)

As \(x\) and \(y\) ranges over all real values, what is the minimum value of

\[ (15x + 30y + 20)^2 + (20x + 40y + 15)^2? \]

Given \(y=\frac{2x}{1+x^{2}}\), where \(x\) and \(y\) are real numbers, what is the range of \(y^{2}+y-2\)?

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