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Polynomial Inequalities

When does revenue exceed cost? When will one race car pass another? Model these values with polynomials and use polynomial inequalities to solve questions like these and more.

Level 3

         

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

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

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

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

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