Graphing Systems of Inequalities

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

But first...

Let's review some previously learned skills that you can apply to this new concept.

Multiple Choice

Graphing Linear Equations: 

Which graph represents the equation $$y=4x-1$$  

Multiple Choice

Systems of Equations:

What is the solution to this system?

• (1, -1)

• (-1, 1)

• (0, -2)

• (2, 0)

Graphing Linear Inequalities

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.

Multiple Choice

How would you graph the following linear inequality?

y ≥ 4

• Dashed line, shade above

• Dashed line, shade below

• Solid line, shade above

• Solid line, shade below

Multiple Choice

How would you graph the following linear inequality?

$$y\ ≤\ \frac{4}{3}x\ -\ 4$$  

• Dashed line, shade above

• Dashed line, shade below

• Solid line, shade above

• Solid line, shade below

Multiple Choice

• y < 2x - 5

• y > 2x - 5

• y ≤ 2x - 5

• y ≥ 2x - 5

Multiple Choice

Graphing Linear Inequalities:

Which point below is NOT a part of the solution set?

• (0, 0)

• (0, 1)

• (-100, -3)

• (5, 10)

Multiple Choice

• y < -4 

• y > -4

• y ≤ -4 

• y ≥ -4 

Multiple Choice

• y < 3/5x - 5

• y > 3/5x - 5

• y ≤ 3/5x - 5

• y ≥ 3/5x - 5

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

Steps to Graphing Systems of Inequalities

• Make sure both inequalities are in Slope-Intercept Form.

• Graph both inequalities (Don't forget to determine dashed or solid line and shade above or shade below).

• Heavily shade in the portion where the shaded regions overlap. This is the solution to the system of inequalities.

Multiple Choice

What is the equation of the graph?

• $$y>2x+1$$  

• $$y>\frac{1}{2}x+1$$  

• $$y<2x+1$$  

• $$y<\frac{1}{2}x+1$$  

Multiple Choice

What is the equation of the graph?

• $$y<2x-4$$  

• $$y<\frac{1}{2}x-4$$  

• $$y>2x-4$$  

• $$y>\frac{1}{2}x-4$$  

​Systems of Inequalities

• ​Graphing systems of inequalities is simply placing two inequalities on the same graph.

• ​The solution is where the shaded areas overlap

• ​If you can graph one inequality, you can graph two of them on the same graph!

Multiple Choice

A solution to system of inequalities will...

• satisfy both inequalities

• satisfy at least one inequality

• not satisfy either inequality.

• Satisfy means make it a true statement.

  4>3 is a true statement.

We will start by graphing.

Multiple Select

To graph:

$$y<3x-2$$

Select all that apply.

• y-intercept = -2
  
  slope = 3

• solid line
  

• dotted line

• shade above

• shade below

Multiple Select

To graph:

$$y\ge x+1$$

Select all that apply.

• y-intercept = 1
  
  slope = 1

• solid line
  

• dotted line

• shade above

• shade below

Let's make them a system!

Graph both inequalities on the same coordinate plane

But wait...where is the solution?!

Poll

Where do you think the solution will be?

• Blue

• Purple

• Red

• White

• All of blue and all of red

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

Open Ended

What makes (1,-1) the solution to the system of equations in the video?

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 of the following is a solution to the systems of inequalities?

• (-1, -3)

• (3, 0)

• (-2, 2)

• (4, 1)

Multiple Choice

Which of the following is a solution to the systems of inequalities?

• (0, 0)

• (1, 1)

• (-4, -2)

• (4, -2)

Multiple Choice

Which of the following is a solution to the systems of inequalities?

• (0, -2)

• (-3, 0)

• (-3, 2)

• (1, 1)

Multiple Choice

Which of the following is NOT a solution to the systems of inequalities?

• (0, 0)

• (3, -2)

• (3, -4)

• (4, -4)

Multiple Choice

Which of the following is a solution to the systems of inequalities?

• (0, 0)

• (-3, 0)

• (2, -5)

• (3, -1)

System of Inequalities

Is made up of 2 or more linear inequalities.

Solution to a System of Inequalities

Is all ordered pairs that make ALL inequalites true.

**WHERE THE SHADED PORTIONS OF THE BOTH INEQUALITIES OVERLAP.**

Multiple Choice

Given the graph, determine which is NOT a solution.

• (0, 0)

• (2, -1)

• (-2, -2)

• (-5, 1)

Multiple Choice

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

• Solution

• Not a solution

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Given the system, determine a solution.

• No solution

• Infinite solutions

• (-5, 5)

• (5, -5)

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

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines
• the same line 
• Intersecting lines

Multiple Choice

What is the solution to this system?

• One Solution
• No solution
• Infinitely Many Solutions

Multiple Choice

Based on the equations for each line, determine how many solutions the system would have.

y=2x+3

y=-4x+3

• Infinite Number of Solutions

• No Solution

• One Solution

Multiple Choice

Based on the equations for each line, determine how many solutions the system would have.

y=5x+3

y=5x-2

• Infinite Number of Solutions

• No Solution

• One Solution

Multiple Choice

Based on the equations for each line, determine how many solutions the system would have.

y=-3x+2

y=-3x+2

• Infinite Number of Solutions

• No Solution

• One Solution

Match

Match the following graphs to the systems of equations

 

 

  

   

• $$y=\frac{2}{3}x+1$$  

  $$y=\frac{2}{3}x-2$$  

• $$y=-x+7$$   $$y=2x+1$$  

• $$y=-2x+4$$   $$y=-2x+4$$