Chapter 5

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x-y \leq 2 \\ x>-2 \\ y \leq 3 \end{array}\right. $$

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

Application Exercises Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 3 x+y \leq 6 \\ x>-2 \\ y \leq 4 \end{array}\right. $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+5 y<10 \\ 3 x+4 y \leq 12 \end{array}\right. $$

Use the formula for the area of a rectangle and the Pythagorean Theorem to solve Exercises \(59-60\) The area of a rug is 108 square feet and the length of its diagonal is 15 feet. Find the length and width of the rug. (PICTURE CANT COPY)

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+y<4 \\ 2 x-3 y \leq 6 \end{array}\right. $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. I apply partial fraction decompositions for rational expressions of the form \(\frac{P(x)}{Q(x)},\) where \(P\) and \(Q\) have no common factors and the degree of \(P\) is greater than the degree of \(Q .\)

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 3 x+y \leq 6 \\ 2 x-y \leq-1 \\ x>-2 \\ y<4 \end{array}\right. $$

Exercises \(61-64\) describe a number of business ventures. For each exercise a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$ Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 2 x+y \leq 6 \\ x+y>2 \\ 1 \leq x \leq 2 \\ y<3 \end{array}\right. $$

Exercises \(61-64\) describe a number of business ventures. For each exercise a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$ \frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3} $$

Write each sentence as an inequality in two variables. Then graph the inequality. The \(y\) -variable is at least 4 more than the product of \(-2\) and the \(x\) -variable.

Exercises \(61-64\) describe a number of business ventures. For each exercise a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125 .\) ( In solving this exercise, let \(x\) represent the number of sold-out performances.)

Writing in Mathematics What is a system of nonlinear equations? Provide an example with your description.

Write each sentence as an inequality in two variables. Then graph the inequality. The \(y\) -variable is at least 2 more than the product of \(-3\) and the \(x\) -variable.

Exercises \(61-64\) describe a number of business ventures. For each exercise a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost \(2 \notin\) per card and you are selling each card for \(50 \phi\). (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

Find the partial fraction decomposition of $$ \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} $$

Writing in Mathematics Explain how to solve a nonlinear system using the substitution method. Use \(x^{2}+y^{2}=9\) and \(2 x-y=3\) to illustrate your explanation.

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is at most \(4 .\) The \(y\) -variable added to the product of 3 and the \(x\) -variable does not exceed 6

will help you prepare for the material covered in the next section. Solve by the substitution method: $$ \left\\{\begin{array}{l} 4 x+3 y=4 \\ y=2 x-7 \end{array}\right. $$

Writing in Mathematics Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}=5\) and \(3 x^{2}-2 y^{2}=19\) to illustrate your explanation.

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is at most \(3 .\) The \(y\) -variable added to the product of 4 and the \(x\) -variable does not exceed 6

Solve by the addition method: $$ \left\\{\begin{array}{l} 2 x+4 y=-4 \\ 3 x+5 y=-3 \end{array}\right. $$

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is no more than 2. The \(y\) -variable is no less than the difference between the square of the \(x\) -variable and 4

will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.

Technology Exercises Write a system of equations, one equation whose graph is a line and the other whose graph is a parabola, that has no ordered pairs that are real numbers in its solution set. Graph the equations using a graphing utility and verify that you are correct.

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the squares of the \(x\) -variable and the \(y\) -variable is no more than \(25 .\) The sum of twice the \(y\) -variable and the \(x\) -variable is no less than 5

determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the same steps to solve nonlinear systems as I did to solve linear systems, although I don't obtain linear equations when a variable is eliminated.

Rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$ \left\\{\begin{array}{l} |x| \leq 2 \\ |y| \leq 3 \end{array}\right. $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a nonlinear system that modeled the orbits of Earth and Mars, and the graphs indicated the system had a solution with a real ordered pair.

Rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$ \left\\{\begin{array}{l} |x| \leq 1 \\ |y| \leq 2 \end{array}\right. $$

Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In 2000 , the cost of Social Security was \(5.48 \%\) of the gross domestic product, increasing by \(0.04 \%\) of the GDP per year. In \(2000,\) the cost of Medicare was \(1.84 \%\) of the gross domestic product, increasing by \(0.17 \%\) of the GDP per year. a. Write a function that models the cost of Social Security as a percentage of the GDP \(x\) years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP \(x\) years after 2000 . c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?

determine whether each statement makes sense or does not make sense, and explain your reasoning. Without using any algebra, it's obvious that the nonlinear system consisting of \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}=25\) does not have real-number solutions.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\)

determine whether each statement makes sense or does not make sense, and explain your reasoning. I think that the nonlinear system consisting of \(x^{2}+y^{2}=36\) and \(y=(x-2)^{2}-3\) is easier to solve graphically than by using the substitution method or the addition method.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\)

Without graphing, Determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l}3 x+y<9 \\\3 x+y>9\end{array}\right.$$

Without graphing, Determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l}6 x-y \leq 24 \\\6 x-y>24\end{array}\right.$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions.

Without graphing, Determine if each system has no solution or infinitely many solutions. $$ \left\\{\begin{array}{l} (x+4)^{2}+(y-3)^{2} \leq 9 \\ (x+4)^{2}+(y-3)^{2} \geq 9 \end{array}\right. $$

Without graphing, Determine if each system has no solution or infinitely many solutions. $$ \left\\{\begin{array}{l} (x-4)^{2}+(y+3)^{2} \leq 24 \\ (x-4)^{2}+(y+3)^{2} \geq 24 \end{array}\right. $$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The points of intersection of the graphs of \(x y=20\) and \(x^{2}+y^{2}=41\) are joined to form a rectangle. Find the area of the rectangle.

A hotel has 200 rooms. Those with kitchen facilities rent for \(\$ 100\) per night and those without kitchen facilities rent for \(\$ 80\) per night. On a night when the hotel was completely occupied, revenues were \(\$ 17,000 .\) How many of each type of room does the hotel have?

Solve the systems in Exercises \(79-80 .\) $$ \left\\{\begin{array}{l} \log _{y} x=3 \\ \log _{y}(4 x)=5 \end{array}\right. $$

A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280 .\) What are the lot's dimensions?

Solve the systems in Exercises \(79-80 .\) $$ \left\\{\begin{array}{l} \log x^{2}=y+3 \\ \log x=y-1 \end{array}\right. $$

A rectangular lot whose perimeter is 320 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 16\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 5\) per foot. The total cost of the fencing along the three sides comes to \(\$ 2140 .\) What are the lot's dimensions?