5.6 Graphing Systems of Linear Inequalities

$$\{\begin{array}{c}3x+y>5\hfill \\ 2x-y\le 10\hfill \end{array}$$

ⓐ $(3,\mathrm{-3})$ ⓑ $(7,1)$

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

ⓐ $(5,\mathrm{-2})$ ⓑ $(\mathrm{-1},3)$

$$\{\begin{array}{c}y>\frac{2}{3}x-5\hfill \\ x+\frac{1}{2}y\le 4\hfill \end{array}$$

ⓐ $(6,\mathrm{-4})$ ⓑ $(3,0)$

$$\{\begin{array}{c}y<\frac{3}{2}x+3\hfill \\ \frac{3}{4}x-2y<5\hfill \end{array}$$

ⓐ $(\mathrm{-4},\mathrm{-1})$ ⓑ $(8,3)$

$$\{\begin{array}{c}7x+2y>14\hfill \\ 5x-y\le 8\hfill \end{array}$$

ⓐ $(2,3)$ ⓑ $(7,\mathrm{-1})$

$$\{\begin{array}{c}6x-5y<20\hfill \\ \mathrm{-2}x+7y>\mathrm{-8}\hfill \end{array}$$

ⓐ $(1,\mathrm{-3})$ ⓑ $(\mathrm{-4},4)$

$$\{\begin{array}{c}2x+3y\ge 2\hfill \\ 4x-6y<\mathrm{-1}\hfill \end{array}$$

ⓐ $\left(\frac{3}{2},\frac{4}{3}\right)$ ⓑ $\left(\frac{1}{4},\frac{7}{6}\right)$

$$\{\begin{array}{c}5x-3y<\mathrm{-2}\hfill \\ 10x+6y>4\hfill \end{array}$$

ⓐ $\left(\frac{1}{5},\frac{2}{3}\right)$ ⓑ $\left(-\frac{3}{10},\frac{7}{6}\right)$

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

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

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

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

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

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

$$\{\begin{array}{c}3x-y\le 6\hfill \\ y\ge -\frac{1}{2}x\hfill \end{array}$$

$$\{\begin{array}{c}2x+4y\ge 8\hfill \\ y\le \frac{3}{4}x\hfill \end{array}$$

$$\{\begin{array}{c}2x-5y<10\hfill \\ 3x+4y\ge 12\hfill \end{array}$$

$$\{\begin{array}{c}3x-2y\le 6\hfill \\ \mathrm{-4}x-2y>8\hfill \end{array}$$

$$\{\begin{array}{c}2x+2y>\mathrm{-4}\hfill \\ -x+3y\ge 9\hfill \end{array}$$

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

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

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

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

$$\{\begin{array}{c}y\le -\frac{2}{3}x+5\hfill \\ x\ge 3\hfill \end{array}$$

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

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

$$\{\begin{array}{c}3x-4y<8\hfill \\ x<1\hfill \end{array}$$

$$\{\begin{array}{c}\mathrm{-3}x+5y>10\hfill \\ x>\mathrm{-1}\hfill \end{array}$$

$$\{\begin{array}{c}x\ge 3\hfill \\ y\le 2\hfill \end{array}$$

$$\{\begin{array}{c}x\le -1\hfill \\ y\ge 3\hfill \end{array}$$

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

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

$$\{\begin{array}{c}\mathrm{-2}x+6y<0\hfill \\ 6y>2x+4\hfill \end{array}$$

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

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

$$\{\begin{array}{c}y\ge \frac{1}{2}x-1\hfill \\ \mathrm{-2}x+4y\ge 4\hfill \end{array}$$

$$\{\begin{array}{c}y\le -\frac{1}{4}x-2\hfill \\ x+4y<6\hfill \end{array}$$

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

$$\{\begin{array}{c}3y>x+2\hfill \\ \mathrm{-2}x+6y>8\hfill \end{array}$$

$$\{\begin{array}{c}y<\frac{3}{4}x-2\hfill \\ \mathrm{-3}x+4y<7\hfill \end{array}$$

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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Will she make a profit if she sells 20 portraits and 35 landscapes?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Can he buy 15 bags of fertilizer and 4 bags of peat moss?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Can she mail 60 cards and 26 packages?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Can he spend 4 hours on Chemistry and 20 hours on Algebra?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Could she buy 5 bananas and 6 granola bars?
4. ⓓ 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

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Could he buy 3 bottles of protein water and 1 protein bar?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?
4. ⓓ 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.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Could he meet his goal by walking 3 miles and running 1 mile?
ⓓ 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.

1. ⓐ Write a system of inequalities to model this problem.
2. ⓑ Graph the system.
3. ⓒ Could the tickets cost $20 for adults and $8 for children?
4. ⓓ 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.

1. ⓐ Write a system of inequalities to model this situation.
2. ⓑ Graph the system.
3. ⓒ Could they take 9 children and 4 adults to both shows?
4. ⓓ Could they take 8 children and 5 adults to both shows?

Graph the inequality $x-y\ge 3$. How do you know which side of the line $x-y=3$ should be shaded?

Graph the system $\{\begin{array}{c}x+2y\le 6\hfill \\ y\ge -\frac{1}{2}x-4\hfill \end{array}$. What does the solution mean?

$$\{\begin{array}{c}x+3y=\mathrm{-9}\hfill \\ 2x-4y=12\hfill \end{array}$$

ⓐ $(\mathrm{-3},\mathrm{-2})$ ⓑ $(0,\mathrm{-3})$

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

ⓐ $\left(6,2\right)$ ⓑ $\left(9,\mathrm{-1}\right)$

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

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

$$\{\begin{array}{c}2x-y=6\hfill \\ y=4\hfill \end{array}$$

$$\{\begin{array}{c}x+4y=\mathrm{-1}\hfill \\ x=3\hfill \end{array}$$

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

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

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

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

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

$$\{\begin{array}{c}y=-\frac{3}{4}x+1\hfill \\ 6x+8y=8\hfill \end{array}$$

LaVelle is making a pitcher of caffe mocha. For each ounce of chocolate syrup, she uses five ounces of coffee. How many ounces of chocolate syrup and how many ounces of coffee does she need to make 48 ounces of caffe mocha?

Eli is making a party mix that contains pretzels and chex. For each cup of pretzels, he uses three cups of chex. How many cups of pretzels and how many cups of chex does he need to make 12 cups of party mix?

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

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

$$\{\begin{array}{c}x-y=0\hfill \\ 2x+5y=\mathrm{-14}\hfill \end{array}$$

$$\{\begin{array}{c}y=\mathrm{-2}x+7\hfill \\ y=\frac{2}{3}x-1\hfill \end{array}$$

$$\{\begin{array}{c}y=\mathrm{-5}x\hfill \\ 5x+y=6\hfill \end{array}$$

$$\{\begin{array}{c}y=-\frac{1}{3}x+2\hfill \\ x+3y=6\hfill \end{array}$$

The sum of two number is 55. One number is 11 less than the other. Find the numbers.

The perimeter of a rectangle is 128. The length is 16 more than the width. Find the length and width.

The measure of one of the small angles of a right triangle is 2 less than 3 times the measure of the other small angle. Find the measure of both angles.

Gabriela works for an insurance company that pays her a salary of $32,000 plus a commission of $100 for each policy she sells. She is considering changing jobs to a company that would pay a salary of $40,000 plus a commission of $80 for each policy sold. How many policies would Gabriela need to sell to make the total pay the same?

$$\{\begin{array}{c}x+y=12\hfill \\ x-y=\mathrm{-10}\hfill \end{array}$$

$$\{\begin{array}{c}4x+2y=2\hfill \\ \mathrm{-4}x-3y=\mathrm{-9}\hfill \end{array}$$

$$\{\begin{array}{c}3x-8y=20\hfill \\ x+3y=1\hfill \end{array}$$

$$\{\begin{array}{c}3x-2y=6\hfill \\ 4x+3y=8\hfill \end{array}$$

$$\{\begin{array}{c}9x+4y=2\hfill \\ 5x+3y=5\hfill \end{array}$$

$$\{\begin{array}{c}-x+3y=8\hfill \\ 2x-6y=\mathrm{-20}\hfill \end{array}$$

The sum of two numbers is $\mathrm{-90}$. Their difference is $16$. Find the numbers.

Omar stops at a donut shop every day on his way to work. Last week he had 8 donuts and 5 cappuccinos, which gave him a total of 3,000 calories. This week he had 6 donuts and 3 cappuccinos, which was a total of 2,160 calories. How many calories are in one donut? How many calories are in one cappuccino?

$$\{\begin{array}{c}6x-5y=27\hfill \\ 3x+10y=\mathrm{-24}\hfill \end{array}$$

$$\{\begin{array}{c}y=3x-9\hfill \\ 4x-5y=23\hfill \end{array}$$

The sum of two numbers is $\mathrm{-32}$. One number is two less than twice the other. Find the numbers.

Four times a number plus three times a second number is $\mathrm{-9}$. Twice the first number plus the second number is three. Find the numbers.

Last month Jim and Debbie earned $7,200. Debbie earned $1,600 more than Jim earned. How much did they each earn?

Henri has $24,000 invested in stocks and bonds. The amount in stocks is $6,000 more than three times the amount in bonds. How much is each investment?

Pam is 3 years older than her sister, Jan. The sum of their ages is 99. Find their ages.

Mollie wants to plant 200 bulbs in her garden. She wantsall irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?

The difference of two supplementary angles is 58 degrees. Find the measures of the angles.

Two angles are complementary. The measure of the larger angle is five more than four times the measure of the smaller angle. Find the measures of both angles.

Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.

The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.

Sheila and Lenore were driving to their grandmother’s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?

Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes $(\frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.

A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

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?