A.5 Systems of Inequalities

Systems of Inequalities

Solutions

A solution is in the area where the shadings overlap

Graphing Systems of Linear Inequalities

Learning Outcomes:

1. Be able to graph a system of linear inequalities

2. Identify solutions to systems of linear inequalities by graphing

Multiple Choice

Systems of Equations:

What is the solution to this system?

• (1, -1)

• (-1, 1)

• (0, -2)

• (2, 0)

Multiple Choice

Graphing Linear Inequalities:

Consider the function y < 2x+3. Which is true?

• The line would be solid with shading above.

• The line would be dashed with shading above.

• The line would be solid with shading below.

• The line would be dashed with shading below.

What is a System of Linear Inequalities?

• A set of at least two linear inequalities in the same variables

• To solve a system, you need to find the variable values that will make each inequality true at the same time

Let's think about systems of equations for a moment...

Open Ended

Which area (white, red, purple, blue) do you think represents the solution to this system of inequalities? Explain your reasoning.

_{Remember that a solution makes BOTH inequalities true at the same time.}

Multiple Choice

Which point is a solution?

• (0, 6)

• (-3, 5)

• (1, 0)

• (-4, 3)

Multiple Choice

Is (0,0) a solution to the system?

• Solution

• Not a solution

Multiple Choice

Which of the following is not a solution to this system of inequalities?

• (0,-1)

• (0,3)

• (4,0)

• (6,-2)

Multiple Choice

Solid or dashed line?  Y <

• Solid

• Dashed

Multiple Choice

Which system of inequalities is shown?

• y ≥ -x - 1 y < x + 4

• y < -x - 1  y ≥ x + 4

• y > -x - 1 y ≥ x + 4

• y > -x - 1 y ≥ x + 3

Multiple Choice

Solid or dashed line? Y ≥

• Solid

• Dashed

Multiple Choice

Which system of inequalities matches the graph?

• y > 2x - 4 y < 2

• y < 2x - 4 y < 2

• y > 2x - 4 y <  2 

• y < 2x - 4 y <   -2

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Graph:
y≤-2x+5
y>2x+7
Which of the following is NOT a solution?

• (-4, 2)

• (-10, 7)

• (-3, 0)

• (-2, 6)

Multiple Choice

Solve the system of inequalities by graphing. $$8x\ +4y\ge10\ and\ 3x-6y>12$$  

Multiple Choice

Identify the graph represented by the following system of inequalities.

2x - y > 4

4x + 2y < 6

Multiple Choice

Is (-2, 0) a solution to the system?

• Solution

• Not a solution

Multiple Choice

Given the graph, determine which is NOT a solution.

• (0, 0)

• (2, -1)

• (-2, -2)

• (-5, 1)

Multiple Choice

Given the system, determine a solution.

• No solution

• Infinite solutions

• (-5, 5)

• (5, -5)

Systems of Inequalities Word Problems

Example:​

When solving inequality word problems, remember the key words and phrases that go along with each inequality symbol:

Multiple Choice

Bobbi can work at most 20 hours next week, but she needs to earn at least $150. Her dog-walking job pays $6 per hour and her job as a car wash attendant pays $10 per hour. 

Let x = # of hours dog-walking

Let y = # of hours at car wash

Which system of inequalities represents the scenario?

• $$6x+10y\ge150$$ $$x+y\le20$$

• $$6x+10y\le150$$ $$x+y\ge20$$

• $$x+y\le150$$ $$6x+10y\ge20$$

• $$x+y\ge150$$ $$6x+10y\le20$$

Multiple Choice

Bobbi can work at most 20 hours next week, but she needs to earn at least $150. Her dog-walking job pays $6 per hour and her job as a car wash attendant pays $10 per hour.

Let x = # of hours dog-walking

Let y = # of hours at car wash

Which graph represents the system of inequalities?

Multiple Choice

Suppose you can spend no more than 15 hours a week at your two jobs. Mowing lawns pays $3 an hour and babysitting pays $5 an hour. You need to earn at least $60 a week.

Let x = # of hours mowing

Let y = # of hours babysitting

Which graph represents the system of inequalities?

Multiple Choice

Kate is selling bracelets and earrings on Etsy to make money for summer vacation. The bracelets (x) cost $2 and earrings (y) cost $3. She needs to make at least $100, but can only make up to 40 pieces of jewelry. Which inequalities represent this situation?

• 2x + 3y > 100

  x + y ≤ 40

• 100x + 2y > 3

  x + y ≤ 40

• 100x + 40y > 100

  3x + 2y ≤ 40

• 3x + 2y > 100

  x + y ≤ 40

Multiple Choice

Luka and Dana are making cookies for the Westland Pantry. They only have up to 6 hours to bake! They can bake 32 chocolate chip cookies in an hour (x), and 20 snickerdoodle cookies in an hour (y). They need to make at least 150 cookies. Which inequalities represent the situation?

• x + y ≥ 6 32x + 20y ≤ 150

• x + y ≥ 632 x + 20y ≥ 150

• x + y ≤ 6 32x + 20y ≤ 150

• x + y ≤ 6 32x + 20y ≥ 150

Multiple Choice

Write the system of inequalities
Carlos works at a movie theater selling tickets. The theater has 300 seats and charges $7.50 for adults and $5.50 for children. The theater expects to make at least $2000 for each showing

• x+y≤300 7.5x+5.5y≥2000

• x+y<300 7.5x+5.5y≥2000

• x+y≤300 7.5x+5.5y≤2000

• x+y>2000 7.5x+5.5y≤300

Multiple Choice

Luka and Dana are making cookies for the Westland Pantry. They only have up to 6 hours to bake! They can bake 32 chocolate chip cookies in an hour (x), and 20 snickerdoodle cookies in an hour (y). They need to make at least 150 cookies. Which inequalities represent the situation?

• x + y ≥ 6 32x + 20y ≤ 150

• x + y ≥ 632 x + 20y ≥ 150

• x + y ≤ 6 32x + 20y ≤ 150

• x + y ≤ 6 32x + 20y ≥ 150

Multiple Choice

Kate is selling bracelets and earrings on Etsy to make money for summer vacation. The bracelets (x) cost $2 and earrings (y) cost $3. She needs to make at least $100, but can only make up to 40 pieces of jewelry. Which inequalities represent this situation?

• 2x + 3y > 100

  x + y ≤ 40

• 100x + 2y > 3

  x + y ≤ 40

• 100x + 40y > 100

  3x + 2y ≤ 40

• 3x + 2y > 100

  x + y ≤ 40

Multiple Choice

Cole is doing a basketball challenge for charity. In order to win money for the charity, he needs to score at least 20 points in less than 10 shots. If x is the number of 2-point shots, and y is the number of three point shots, which inequalities below represent the situation?

• x + y < 10

  2x + 3y ≥ 20

• x + y ≤ 10

  2x + 3y ≥ 20

• x + y ≤ 10

  2x + 3y > 20

• x + y < 10

  2x + 3y > 20

Fill in the Blank

The hole for a birdfeeder post is 3 feet deep. The top of the post needs to be at least 5 feet above the ground.

What is the key word or words that would tell you what inequality symbol to use to solve this problem?

Multiple Choice

JoAnn likes her job as a baby-sitter, but it pays only $3 per hour. She has been offered a job as a tutor that pays $6 per hour. Because of school, her parents only allow her to work a maximum of 15 hours per week. How many hours can JoAnn tutor and baby-sit and still make at least $66 per week? How much does she make?

• b + t < 15 3b + 6t > 66

• b + t > 15 3b + 6t < 66

• 3b + 6t > 15 b + t < 66

• 3b + 6t < 15 b + t < 66