Chapter 5

Mark wants to invest \(\$ 10,000\) to pay for his daughter's wedding next year. He will invest some of the money in a short term CD that pays \(12 \%\) interest and the rest in a money market savings account that pays \(5 \%\) interest. How much should he invest at each rate if he wants to earn \(\$ 1,095\) in interest in one year?

A trust fund worth \(\$ 25,000\) is invested in two different portfolios. This year, one portfolio is expected to earn \(5.25 \%\) interest and the other is expected to earn \(4 \%\). Plans are for the total interest on the fund to be \(\$ 1150\) in one year. How much money should be invested at each rate?

A business has two loans totaling \(\$ 85,000\). One loan has a rate of \(6 \%\) and the other has a rate of \(4.5 \%\). This year, the business expects to pay \(\$ 4650\) in interest on the two loans. How much is each loan?

Laurie was completing the treasurer's report for her son's Boy Scout troop at the end of the school year. She didn't remember how many boys had paid the \(\$ 15\) full-year registration fee and how many had paid the \(\$ 10\) partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If \(\$ 250\) was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?

As the treasurer of her daughter's Girl Scout troop, Laney collected money for some girls and adults to go to a three-day camp. Each girl paid \(\$ 75\) and each adult paid \(\$ 30\). The total amount of money collected for camp was \(\$ 765 .\) If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}3 x+y>5 \\ 2 x-y \leq 10\end{array}\right.\) (a) (3,-3) (b) (7,1)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}4 x-y<10 \\ -2 x+2 y>-8\end{array}\right.\) (a) (5,-2) (b) (-1,3)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}y>\frac{2}{3} x-5 \\ x+\frac{1}{2} y \leq 4\end{array}\right.\) (a) (6,-4) (b) (3,0)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}y<\frac{3}{2} x+3 \\ \frac{3}{4} x-2 y<5\end{array}\right.\) (a) (-4,-1) (b) (8,3)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}7 x+2 y>14 \\ 5 x-y \leq 8\end{array}\right.\) (a) (2,3) (b) (7,-1)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}6 x-5 y<20 \\ -2 x+7 y>-8\end{array}\right.\) (a) (1,-3) (b) (-4,4)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}2 x+3 y \geq 2 \\ 4 x-6 y<-1\end{array}\right.\) (a) \(\left(\frac{3}{2}, \frac{4}{3}\right)\) (b) \(\left(\frac{1}{4}, \frac{7}{6}\right)\)

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}5 x-3 y<-2 \\ 10 x+6 y>4\end{array}\right.\) (a) \(\left(\frac{1}{5}, \frac{2}{3}\right)\) (b) \(\left(-\frac{3}{10}, \frac{7}{6}\right)\)

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y<-2 x+2 \\ y \geq-x-1 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y<2 x-1 \\ y \leq-\frac{1}{2} x+4 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq-\frac{2}{3} x+2 \\ y>2 x-3 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x-y>1 \\ y<-\frac{1}{4} x+3 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x+2 y<4 \\ y

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} 3 x-y \leq 6 \\ y \geq-\frac{1}{2} x \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} 2 x+2 y>-4 \\ -x+3 y \geq 9 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} 2 x+y>-6 \\ -x+2 y \geq-4 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x-2 y<3 \\ y \leq 1 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x-3 y>4 \\ y \leq-1 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq-\frac{1}{2} x-3 \\ x \leq 2 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \leq-\frac{2}{3} x+5 \\ x \geq 3 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq \frac{3}{4} x-2 \\ y<2 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \leq-\frac{1}{2} x+3 \\ y<1 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} -3 x+5 y>10 \\ x>-1 \end{array}\right. $$

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x \geq 3 \\ y \leq 2 \end{array}\right. $$

$$ \left\\{\begin{array}{l} 2 x+4 y>4 \\ y \leq-\frac{1}{2} x-2 \end{array}\right. $$

$$ \left\\{\begin{array}{l} x-3 y \geq 6 \\ y>\frac{1}{3} x+1 \end{array}\right. $$

$$ \left\\{\begin{array}{l} y \geq 3 x-1 \\ -3 x+y>-4 \end{array}\right. $$

$$ \left\\{\begin{array}{l} y<\frac{3}{4} x-2 \\ -3 x+4 y<7 \end{array}\right. $$

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for \(\$ 15\) and the landscapes for \(\$ 10\). She needs to sell at least \(\$ 800\) worth of drawings in order to earn a profit. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Will she make a profit if she sells 20 portraits and 35 landscapes? (d) Will she make a profit if she sells 50 portraits and 20 landscapes?

Jake does not want to spend more than \(\$ 50\) on bags of fertilizer and peat moss for his garden. Fertilizer costs \(\$ 2\) a bag and peat moss costs \(\$ 5\) a bag. Jake's van can hold at most 20 bags. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Can he buy 15 bags of fertilizer and 4 bags of peat moss? (d) Can he buy 10 bags of fertilizer and 10 bags of peat moss?

Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than \(\$ 500\). The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is \(\$ 3\) and for a package the cost is \(\$ 7\) (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Can she mail 60 cards and 26 packages? (a) Can she mail 90 cards and 40 packages?

Juan is studying for his final exams in Chemistry and Algebra. He knows he only has 24 hours to study, and it will take him at least three times as long to study for Algebra than Chemistry. (a) Write a system of inequalities to model this situation. (b) Graph the system. c) Can he spend 4 hours on Chemistry and 20 hours on Algebra? (d) Can he spend 6 hours on Chemistry and 18 hours on Algebra?

Jocelyn is pregnant and needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of \(\$ 15\) for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost \(\$ 0.35\) each and the granola bars cost \(\$ 2.50\) each. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could she buy 5 bananas and 6 granola bars? (d) Could she buy 3 bananas and 4 granola bars?

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs \(\$ 3.20\) and a protein bar costs \(\$ 1.75 .\) The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has \(\$ 10\) dollars to spend (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could he buy 3 bottles of protein water and 1 protein bar? (a) Could he buy no bottles of protein water and 5 protein bars?

Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese? (a) Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?

Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1,500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could he meet his goal by walking 3 miles and running 1 mile? (d) Could he meet his goal by walking 2 miles and running 2 mile?

Tickets for an American Baseball League game for 3 adults and 3 children cost less than \(\$ 75,\) while tickets for 2 adults and 4 children cost less than \(\$ 62\). (a) Write a system of inequalities to model this problem. (b) Graph the system. (c) Could the tickets cost \(\$ 20\) for adults and \(\$ 8\) for children? (a) Could the tickets cost \(\$ 15\) for adults and \(\$ 5\) for children?

Grandpa and Grandma are treating their family to the movies. Matinee tickets cost \(\$ 4\) per child and \(\$ 4\) per adult. Evening tickets cost \(\$ 6\) per child and \(\$ 8\) per adult. They plan on spending no more than \(\$ 80\) on the matinee tickets and no more than \(\$ 100\) on the evening tickets. (a) Write a system of inequalities to model this situation. (b) Graph the system. c) Could they take 9 children and 4 adults to both shows? (a) Could they take 8 children and 5 adults to both shows?