91Ó°ÊÓ

Graph each inequality. $$3 x-6 y \leq 12$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints \(z=5 x+6 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ 2 x+y \geq 10 \\ x+2 y \geq 10 \\ x+y \leq 10\end{array}\right.\)

write the partial fraction decomposition of each rational expression. $$ \frac{2 x^{2}-18 x-12}{x^{3}-4 x} $$

Graph each inequality. $$y

write the partial fraction decomposition of each rational expression. $$ \frac{6 x-11}{(x-1)^{2}} $$

In Exercises 23-24, let \(x\) represent the first number, \(y\) the second number, and \(z\) the third number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of three numbers is 16. The sum of twice the first number, 3 times the second number, and 4 times the third number is 46. The difference between 5 times the first number and the second number is \(31 .\) Find the three numbers.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I need to be able to graph systems of linear inequalities in order to solve linear programming problems.

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{cc} \hline x, \text { seconds after the ball is } & y, \text { ball's height, in feet, above } \\ \text { thrown } & \text { the ground } \\ \hline 1 & 224 \\ 3 & 176 \\ 4 & 104 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

write the partial fraction decomposition of each rational expression. $$ \frac{9 x+2}{(x-2)\left(x^{2}+2 x+2\right)} $$

A mathematical model can be used to describe the relationship between the number of feet a car travels once the brakes are applied, \(y,\) and the number of seconds the car is in motion after the brakes are applied, \(x .\) A research firm collects the following data: $$\begin{array}{cc} \hline \begin{array}{c} x \text { , seconds in motion } \\ \text { after brakes are applied } \end{array} & \begin{array}{c} y, \text { feet car travels } \\ \text { once the brakes are applied } \end{array} \\ \hline 1 & 46 \\ 2 & 84 \\ 3 & 114 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=6 .\) Describe what this means.