91Ó°ÊÓ

Determine whether the given rational expression is proper or improper. If the expression is improper, rewrite it as the sum of a polynomial and a proper rational expression. $$ \frac{2 x\left(x^{2}+4\right)}{x^{2}+1} $$

Graph each inequality. $$x^{2}+y^{2} \leq 9$$

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ CB

A banquet hall offers two types of tables for rent: 6 -person rectangular tables at a cost of \(\$ 28\) each and 10 -person round tables at a cost of \(\$ 52\) each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The hall can have a maximum of 35 tables, and the hall has only 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{array}{l} x y=1 \\ y=2 x+1 \end{array}\right. $$

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ \(C A+5 I_{3}\)

Graph each system of linear inequalities. $$\left\\{\begin{array}{r}3 x-y \geq 6 \\\x+2 y \leq 2\end{array}\right.$$

An entrepreneur is having a design group produce at least six samples of a new kind of fastener that he wants to market. It costs \(\$ 9.00\) to produce each metal fastener and \(\$ 4.00\) to produce each plastic fastener. He wants to have at least two of each version of the fastener and needs to have all the samples 24 hours (h) from now. It takes 4 h to produce each metal sample and 2 h to produce each plastic sample. To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost of the samples?

Solve each system. Use any method you wish. $$ \left\\{\begin{array}{r} 2 x^{2}-x y+y^{2}=8 \\ x y=4 \end{array}\right. $$

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use \(x, y ;\) or \(x, y, z ;\) or \(x_{1}, x_{2}, x_{3}, x_{4}\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $$ \left[\begin{array}{llll|l} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \end{array}\right] $$