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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.

Solution Sets

         

\( A\) and \( B \) are two subsets of \(\mathbb{R}\) defined by: \[\begin{align} A &= \{ x \mid x^2 - 6 x - 91 \leq 0 \}, \\ B &= \{ x \mid x^2 - 6 x - 91 = 0 \}. \end{align} \] How many integers does the set \( A \cap B^C \) contain?

Details and assumptions

\( B^C \) denotes the complement of set \( B \).

\(\mathbb{R}\) denotes the set of all real numbers.

Sets \(A, B, C\) and \(D\) are defined as follows: \[ \begin{align} A &= \{ (x,y) \vert x>0 \} \\ B &= \{ (x,y) \vert y>0 \} \\ C &= \{ (x,y) \vert y>5x \} \\ D &= \{ (x,y) \vert y<-3x \}.
\end{align} \] Then which of the following is the set that represents the shaded region, including the two axes, in the above graph?

The sets \( A = \{ x \mid x^2 - 5 x + 4 \leq 0 \} \) and \( B = \{ x \mid x^2 + ax + b < 0 \} \) satisfy the following conditions: \[ A \cap B = \emptyset \mbox{ and } A \cup B = \{ x \mid 1 \leq x < 15 \}. \] What is the value of \( a + b? \)

\( P,\ Q \) and \( R \) are three subsets of \(\mathbb{R}\) defined as follows: \[\begin{align} P &= \{ x \mid x^2 - 21x + 110 > 0 \}, \\ Q &= \{ x \mid x^2 - 15x + 44 < 0 \}, \\ R &= \{ x \mid x^2 - 11ax + 10a^2 < 0,\ a > 0 \}. \end{align} \] What is the sum of the minimum and maximum values of \(a\) such that \( (P \cap Q) \subset R \)?

Details and assumptions

\(\mathbb{R}\) denotes the set of all real numbers.

\( A\), \(B \) and \( C \) are subsets of \(\mathbb{R}\) defined as follows: \[\begin{align} A &= \{ x \mid x^2 - 10 x + 21 \geq 0 \}, \\ B &= \{ x \mid x^2 - 10 x + 21 > 0 \},\\ C &= \{ x \mid x^2 - 19 x + 48 = 0 \}. \end{align} \] What is the value of the sum of all the elements of the set \( (A \cap B^C) \cup C \)?

Details and assumptions

\( B^C \) denotes the complement of set \( B \).

\(\mathbb{R}\) denotes the set of all real numbers.

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