Nonlinear Inequalities

Systems of inequalities and word problems, nonlinear systems

By Shalynne Orth

Linear Inequalities

We will be able to match a linear equality with the graph and find solutions to linear inequalities.

I will be able to participate in the Quizizz lesson.

Finding solutions to linear inequalities

• Remember any point in the shaded region is a solution and if it is a solid line then any of those points on the line are solutions.

• Any point that is NOT in the shaded region is not a solution and if it is a dashed line then any point that is on the line is a not solution.

Multiple Choice

Shade above or below?

y <

• Above

• Below

Multiple Choice

Shade above or below?
y ≥

• Above
• Below

Multiple Choice

Solid or dashed line?  
Y <

• Solid
• Dashed

Multiple Choice

Are the points on the line part of the solution set or not?

• Yes, they are part of the solution set

• No, they are NOT part of the solution set

Multiple Choice

Choose the correct inequality for this graph.

Multiple Choice

Which of the following inequalities match the given graph?

• y ≤ -5/4 x + 3

• y ≥ -5/4 x + 3

• y < -5/4 x + 3

• y > -5/4 x + 3

Multiple Choice

Is the point (3, 1) a solution for the following inequality?

• Yes because it is located in the shaded region.

• No because it is located on the line and the line is dotted.

• Yes because it is located on the line.

• No because it is in the shaded region.

Multiple Choice

Which inequality matches the graph?

• y < 5x + 1

• y > 5x + 1

• y ≤ 5x + 1

• y ≥ 5x + 1

Systems of Inequalities

Solutions to the System

must be in the DOUBLE SHADED area

where the shading for the two lines overlaps

Multiple Choice

Is (-2,4) a solution to the system?

• Solution
• Not a solution

Multiple Choice

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

• Solution
• Not a solution

Multiple Choice

Which point is a solution?

• (0, 6)
• (-3, 4)
• (1, 0)
• (-4, 3)

Multiple Choice

Match the system to the graph (solution set is on the left)

• y ≤ 3x   and  

  y ≤ -2x + 4

• y > 3x   and  

  y > -2x + 4

• y < 3x  and

   y > -2x + 3

• y > 3x   and  

  y < -2x + 4

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

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

Which graph represents the solution of this systems of inequalities?

$$x-3y\le-6$$  

$$2x+y>6$$  

Multiple Choice

This system has an infinite number of solutions, TRUE or FALSE?

• TRUE
• FALSE

Multiple Choice

Which of the following points is a solution to the system of equations below?
x + 2y ≥ 4
3x − 2y < 4
Take your time!

• (0, 0)
• (2, 1)
• (2, 2)
• (-1, 1)

Multiple Choice

What's the difference between systems of equations and systems of inequalities?

• Systems of equations have 1 solution, no solution, or infinite solutions while systems of inequalities can have a range of solutions.

• Systems of equations use slope-intercept form only while systems of inequalities use standard form only.

• Systems of equations have a shaded region while systems of inequalities do not.

• There is no difference.

Systems of Inequalities Word Problems

Example:​

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

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

Nonlinear Systems: Solve by Graphing

• Graph both equations

• Solutions are the intersection points (just like with linear systems!)

• To check your answer, plug it in and make sure it makes both equations true!​

Multiple Choice

• A
• B
• C
• D

Multiple Choice

Identify the solution to the system graphed here.

• (-4, 0) and (1, 0)

• (-4, -1)

• (-4, 1)

• No solution

Multiple Choice

 Determine if the point (2, 0) is a solution to the following system: $$y=2x-4$$ $$y=x^2-4$$  

• Yes, because it works in only one equation.

• Yes, because it works in both equations.

• No, because it works in neither equation.

• No, because it works in only one equation.

Multiple Choice

Which of the following graphs correctly represents: $$y=x^2$$   $$y=-2x+1$$  

Multiple Choice

Identify the solution to the system graphed here.

• (0, 4) and (0, -2)

• (2, -2)

• (0 4)

• No solution

Multiple Choice

Identify the solution to the system graphed here.

• (1, 5) and (-4, 0)

• (1, 5) and (-2, 0)

• (1, 5) and (0, -4)

• No solution