Algebra

Polynomial Inequalities

Polynomial Inequalities - Sets

         

A A and B B are two subsets of R\mathbb{R} defined by: A={xx26x910},B={xx26x91=0}.\begin{aligned} A &= \{ x \mid x^2 - 6 x - 91 \leq 0 \}, \\ B &= \{ x \mid x^2 - 6 x - 91 = 0 \}. \end{aligned} How many integers does the set ABC A \cap B^C contain?

Details and assumptions

BC B^C denotes the complement of set B B .

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

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

The sets A={xx25x+40} A = \{ x \mid x^2 - 5 x + 4 \leq 0 \} and B={xx2+ax+b<0} B = \{ x \mid x^2 + ax + b < 0 \} satisfy the following conditions: AB= and AB={x1x<15}. A \cap B = \emptyset \mbox{ and } A \cup B = \{ x \mid 1 \leq x < 15 \}. What is the value of a+b? a + b?

P, Q P,\ Q and R R are three subsets of R\mathbb{R} defined as follows: P={xx221x+110>0},Q={xx215x+44<0},R={xx211ax+10a2<0, a>0}.\begin{aligned} 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{aligned} What is the sum of the minimum and maximum values of aa such that (PQ)R (P \cap Q) \subset R ?

Details and assumptions

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

A A, BB and C C are subsets of R\mathbb{R} defined as follows: A={xx210x+210},B={xx210x+21>0},C={xx219x+48=0}.\begin{aligned} 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{aligned} What is the value of the sum of all the elements of the set (ABC)C (A \cap B^C) \cup C ?

Details and assumptions

BC B^C denotes the complement of set B B .

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

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