5.6 Graphing systems of linear inequalities

Lynn paid a total of $2,780 for 261 tickets to the theater. Student tickets cost $10 and adult tickets cost $15. How many student tickets and how many adult tickets did Lynn buy?

Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?

Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her $1.29 per cup. The candies are $2.49 per cup and the nuts are $0.69 per cup. How many cups of candies and how many cups of nuts should she use?

A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

Jack has $12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?

When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1585. What is the amount of each loan?

$\{\begin{array}{c}4x+y>6\hfill \\ 3x-y\le 12\hfill \end{array}$

ⓐ $(2,\mathrm{-1})$ ⓑ $(3,\mathrm{-2})$

$\{\begin{array}{c}y>\frac{1}{3}x+2\hfill \\ x-\frac{1}{4}y\le 10\hfill \end{array}$

ⓐ $(6,5)$ ⓑ $(15,8)$

$\{\begin{array}{c}y<3x+1\hfill \\ y\ge -x-2\hfill \end{array}$

$\{\begin{array}{c}x-y>\mathrm{-1}\hfill \\ y<\frac{1}{3}x-2\hfill \end{array}$

$\{\begin{array}{c}2x-3y<6\hfill \\ 3x+4y\ge 12\hfill \end{array}$

$\{\begin{array}{c}y\le -\frac{3}{4}x+1\hfill \\ x\ge \mathrm{-5}\hfill \end{array}$

$\{\begin{array}{c}x+3y<5\hfill \\ y\ge -\frac{1}{3}x+6\hfill \end{array}$

$\{\begin{array}{c}y\ge 2x-5\hfill \\ \mathrm{-6}x+3y>\mathrm{-4}\hfill \end{array}$

Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for $12 each and the necklaces for $18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least $500 worth in order to earn a profit.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Should she display 26 bracelets and 14 necklaces?
4. ⓓ Should she display 39 bracelets and 1 necklace?

Annie has a budget of $600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost $4 each and hardcover books cost $15 each.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Can she buy 8 paperback books and 40 hardcover books?
4. ⓓ Can she buy 10 paperback books and 37 hardcover books?

$\{\begin{array}{c}x-4y=\mathrm{-8}\hfill \\ 2x+5y=10\hfill \end{array}$

ⓐ $(0,2)$ ⓑ $(4,3)$

$\{\begin{array}{c}x-y=5\hfill \\ x+2y=\mathrm{-4}\hfill \end{array}$

$\{\begin{array}{c}x-y>\mathrm{-2}\hfill \\ y\le 3x+1\hfill \end{array}$

$\{\begin{array}{c}3x-2y=3\hfill \\ y=2x-1\hfill \end{array}$

$\{\begin{array}{c}x+y=\mathrm{-3}\hfill \\ x-y=11\hfill \end{array}$

$\{\begin{array}{c}4x-3y=7\hfill \\ 5x-2y=0\hfill \end{array}$

$\{\begin{array}{c}y=-\frac{4}{5}x+1\hfill \\ 8x+10y=10\hfill \end{array}$

$\{\begin{array}{c}2x+3y=12\hfill \\ \mathrm{-4}x+6y=\mathrm{-16}\hfill \end{array}$

The sum of two numbers is −24. One number is 104 less than the other. Find the numbers.

Ramon wants to plant cucumbers and tomatoes in his garden. He has room for 16 plants, and he wants to plant three times as many cucumbers as tomatoes. How many cucumbers and how many tomatoes should he plant?

Two angles are complementary. The measure of the larger angle is six more than twice the measure of the smaller angle. Find the measures of both angles.

On Monday, Lance ran for 30 minutes and swam for 20 minutes. His fitness app told him he had burned 610 calories. On Wednesday, the fitness app told him he burned 695 calories when he ran for 25 minutes and swam for 40 minutes. How many calories did he burn for one minute of running? How many calories did he burn for one minute of swimming?

Kathy left home to walk to the mall, walking quickly at a rate of 4 miles per hour. Her sister Abby left home 15 minutes later and rode her bike to the mall at a rate of 10 miles per hour. How long will it take Abby to catch up to Kathy?

It takes $5\frac{1}{2}$ hours for a jet to fly 2,475 miles with a headwind from San Jose, California to Lihue, Hawaii. The return flight from Lihue to San Jose with a tailwind, takes 5 hours. Find the speed of the jet in still air and the speed of the wind.

Liz paid $160 for 28 tickets to take the Brownie troop to the science museum. Children’s tickets cost $5 and adult tickets cost $9. How many children’s tickets and how many adult tickets did Liz buy?

A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?

Translate to a system of inequalities and solve.

Andi wants to spend no more than $50 on Halloween treats. She wants to buy candy bars that cost $1 each and lollipops that cost $0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Can she buy 20 candy bars and 70 lollipops?
4. ⓓ Can she buy 15 candy bars and 65 lollipops?