\documentstyle[12pt,fleqn]{article} \pagestyle{empty} \setlength{\topmargin}{-1in} \setlength{\oddsidemargin}{-.3in} \setlength{\textheight}{9in} \setlength{\textwidth}{7.2in} \begin{document} \begin{enumerate} \item Solve the equation \[ \frac{3x-5}{2}= x+6 \] \item A clerk at a travel agent's can handle in 5 minutes a customer who just demands information, and in 12 minutes a customer who buys a ticket. It is estimated that $70\%$ of the customers just want information, and $30\%$ buy tickets. How many customers can a clerk handle in a 8--hour workday? \item Solve the system of inequalities \[ 5< 2x-8 \le 30-x \] \item Sketch the graph of the equation and find its intersection with the $x$--axis. \[ 3x-2y+3=0 \] \item A straight line goes through the points $(-1,4)$ and $(2,2)$. What is its slope? \item Compute \[ 8x^2-2x+4-(x-1)(x^2+2x+3) \] \item Factor the polynomial $p(x)= 2x^2+12x-14$, and solve the equation $p(x)=0$. \item Compute and simplfy \[ \frac{16-x}{x^2-2x-8}+\frac{4}{x+2}-\frac{2}{x-4} \] \item Factor the polynomial $x^3-4x^2-11x+30$. \item Solve the equation \[ \frac{x+8}{x-2}=3 \] \item Simplify the expression \[ \left.^3 \sqrt{48y^4} \right. (64y^2)^{-\frac{1}{6}} \] \item Find the points in the line $\{ y=3x-1 \}$ such that their distance to $(0,0)$ is 4. \item Find the complex solutions $z_1,z_2$ of the equation \[ z^2+3z+\frac{25}{4}=0 \] and compute $\frac{1}{z_1}+ \frac{1}{z_2}$. \item Find all real numbers $b$ such that the equation \[ 3x^2+ bx-6=0 \] has exactly one solution. \item A plane nose--dives from a height of 6,000 ft. While it does so, its height is given by the equation \[ h(t)= 6,000 - 300t - 16t^2 \] with $h$ the height in feet, $t$ the time in seconds. The plane must straighten its flight at a height of 2,000 ft. For how many seconds can it nose--dive? \item We have to build a rectangular playground such that its width is 20 ft less than its length, and its total area is 4,000 square feet. What should be its length and width? \item Find the equation of the line that goes through the points $(0,4)$ and $(2,-1)$. Compute the intersection of this line with the horizontal line $y=1$. \item Plot the graphs of the line with equation $2x-2y+14=0$ and the parabolla $y=x^2-3x+2$. Compute their intersection points. \item Solve the system \[ \left\{ \begin{array}{cccc} x & -2y & +3z & =0 \\ -x & -y & +z & =2 \\ 2x & & -3z & =1 \end{array} \right. \] \item A 20--oz bag of cookies costs $\$2.20$ to produce, and a 12--oz bag costs $\$1.40$. If the bags that we use in the 2 sizes cost the same, what is the cost of the bag and what is the cost of every ounce of cookies? \item An 80--lb sack of fertilizer has a volume of 0.32 cu ft. We know that it contains a mix of two substances: fertilizer A, with density $300 \frac{\mbox{lb}}{\mbox{cu ft}}$, and fertilizer B, with density $200 \frac{\mbox{lb}}{\mbox{cu ft}}$. Find how much weight of every type of fertilizer does the sack contain. \item A calculator seeling for $\$179.00$ cost the retailer $\$97.00$. Find the markup rate. \item Determine the number of gallons of a $20\%$ alcohal solution and the number of gallons of a $45\%$ solution that are required to make 10 gallons of a $35\%$ solution. \item If $g(x)=x(3-x)$ evaluate and simplify: $g(5)$, $g(-2)$, $g(t+2)$. \item Find the time for 2 people working together to complete a task if it takes them 12 and 8 hours to complete the task working individually. \item Find the slope of the line through $(2,-4)$ and $(-10,1)$, and find the distance between these points. \item Graph $f(x)=4-x^2$ and $h(x)=(x+2)^2$ on the same axes. \item Use elimination to solve: \[ \begin{array}{ccccc} 2x & - & 5y & = & 18 \\ -3x & + & 4y & = & -20 \\ \end{array} \] \item Simplify and factor: \[ 3y^2-(y-3(y^2-5)) \] \item Factor: \[ 64x^3-27 \] \item Simplify: \[ \frac {\frac{2t^2+t-15}{t^2-9}} {\frac{4t^2-20t+25}{t^2-t-6}} \] \item Simplify: \[ \frac {\frac{1}{x}-\frac{1}{y}} {\frac{1}{x^2}-\frac{1}{y^2}} \] \item Divide: \[ \frac{x^4+3x^2-35}{x+2} \] \item Solve: \[ \frac{7}{x+3}-\frac{6}{x-2}=6 \] \item Rewrite using only positive exponents and simplify: \[ (3x^2y^{-1})^2(4x^4y^{-3})^{-\frac{1}{2}} \] \item Rationalize the denominator and simplify: \[ \sqrt[3]{ \frac{32v^7}{9u^2} } \] \item Rationalize the denominator and simplify: \[ \frac{\sqrt{12}}{\sqrt{6}-3} \] \item Write the complex number in standard form $(a+bi)$: \[ \frac{2-\sqrt{-9}}{1+2i} \] \item Solve: \[ 2x^2-6x+7 \] \item A little league baseball team obtains a block of tickets to a ball game for \$96. When three more people decide to go to the game, the price per ticket is decreased by \$1.60. How many people are going to the game? \item Fid the vertex, $x$ and $y$- intercepts and graph: \[ y=2x^2-12x+14 \] \item (10 points) Solve and graph the inequality. \[ -3 < \frac{2x-3}{2} \leq 3 \] \item (10 points) Solve the equation: \[ |x+2|=|3x-1| \] \item (10 points) Find the distance between the points: (1,3) and (-2,7). \item (10 points) Graph the equation, label all $x$ and $y$-intercepts. \[ x-2y-2=0 \] \item Factor completely: \[ 4t^2-24t-13 \] \item The length of a rectangle is $\frac{3}{2}$ times the width. The area of the rectangle is 54 cm$^2$. Find the length and width of the rectangle. \item (15 points) Divide and simplify: \[ \frac {\left( \frac{5x}{x+7} \right)} {\left(\frac{10}{x^2+8x+7}\right)} \] \item (15 points) Simplify \[ \frac {\left( \frac{x}{x-3} - \frac{2}{3} \right)} {\left( \frac{10}{3x} + \frac{x^2}{x-3} \right)} \] \item (15 points) Solve the equation: \[ \frac{3}{x-1} + \frac{6}{x^2-3x+2} = 2 \] \item (20 points) Factor the polynomial. (Hint: $(x-2)$ is a factor) \[ x^3-5x^2-4x+20 \] \item (15 points) Divide: \[ \frac{2x^3-5x^2+x-7}{x-2} \] \item (10 points) Simplify, write your answer using only positive exponents. \[ \left( \frac{2x^2}{y^{-1}} \right) ^{-2} \] \item (15 points) Rationalize the denominator. \[ \frac{6}{7-\sqrt{7}} \] \item (15 points) Solve the equation. \[ -6=\sqrt{x}-x \] \item (15 points) Write the complex number in standard form ($a+bi$). \[ \frac{1+i}{1-i} \] \item (15 points) Find all solutions to the equation: \[ x^{\frac{2}{5}}-x^{\frac{1}{5}}-2=0 \] \item (20 points) Solve the equation: \[ y^2-3y+4=0 \] \item (15 points) You take a 10 mile hike. For the first 6 miles you travel at one speed. Then you pick up the pace and finish the next 4 miles 2 miles per hour faster then the first 6 miles. The trip takes 4 hours. How fast did you go in each part of the hike? \item (15 points) Find the equation of the line through (1,1) and (5,-3). \item (15 points) Graph the inequality: \[ 2x+3y < 9 \] \end{enumerate} \end{document}