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Edsc 304 Interactive Lecture
11
Graphing Systems of Linear Inequalities
Today's Lesson
Reasoning with Equations and Inequalities
Algebra 1 Grade 7
By Julia Reinhart
Class Objectives
Class Objectives
- Graph inequalities and systems of inequalities
- Learn how the solution set is displayed by each graph
- Know where to shade and when to use a dashed or solid line in each graph
- Standards Covered: A-REI-10 and A-REI-12
Review
- How did we graph an equation?
- How did we graph a system of equations?
- What was the solution set for a system of equations?
- y = mx + b
- What happens when this is changed to y < mx + b?
- Where is an example of this in our daily lives?
A Real Life Example
A Real Life Example
Graphing an Equation
ex) y = 2x + 1
Graphing an Equation
Graphing an Inequality
ex) y < 2x + 1
Graphing an Inequality
Graphing a System of Equations
ex) y = 2x + 1 and y = -2x + 1
- Solution is an ordered pair (x, y)
Graphing a System of Equations
Graphing a System of Inequalities
ex) y < 2x + 1 and y > -2x + 1
- Solution is a region (shaded area)
Graphing a System of Inequalities
Steps to Graph an Inequality
1. Graph as if it was an equation.
2. Make a dotted or solid line.
3. Shade accordingly.
ex) Graph y + 1 > 3x
Graph an Inequality
Step 1: Graph as if it was an equation
Step 1
- Solve for y by itself
- Graph using slope and y intercept
ex) y + 1 > 3x
-1 -1
y > 3x - 1
- Slope = m = 3
- Y-intercept = b = -1 -> Start at (0, b) = (0, -1)
- Rise/Run -> 3/1 -> up 3 right 1 or down 3 left 1
Step 2: Make a dotted or solid line
- < or > : Dotted line
- less than or greater than
- the line does not satisfy our inequality
- ≤ or ≥ : Solid line
- less than or equal to, greater than or equal to
- the line does satisfy our inequality
Step 2
ex) y > 3x - 1
- > so dotted line
Step 3: Shade
- >, ≥: greater than, greater than or equal to shade above
- <, ≤: less than, less than or equal to shade below
ex) y > 3x - 1
- > so shade above
Step 3
Chart
Now, you try!
- What will the graph be for the following inequalities?
- Will there be a solid or dotted line? Why?
- Will shading be above or below? Why?
- How does this compare to graphing an equation?
- Which requires more steps graphing an equation or inequality?
Now, you try!
1. (5/3)y + 11 ≤ (5/3)x + 38/3
2. -2y + 30 > -10x + 26
Problem 1
1. (5/3)y + 11 ≤ (5/3)x + 38/3
Multiply both sides by 3 to get rid of the fraction:
3[(5/3)y + 11 ≤ (5/3)x + 38/3]
5y + 33 ≤ 5x + 38
Isolate y.
5y + 33 ≤ 5x + 38
- 33 - 33
(1/5)[5y ≤ 5x + 5]
y ≤ x + 1 (y = mx + b form)
- slope = 1, b = 1 -> y-int = (0, 1)
- ≤ so solid line and shade below
Problem 1
Problem 2
2. -2y + 30 > -10x + 26
Get y by itself.
-2y + 30 > -10x + 26
- 30 - 30
-2y > -10x - 4
Multiply both sides by (-1/2). Because we are multiplying by -1, flip the inequality.
(-1/2)[-2y > -10x - 4]
y < 5x + 2 (y = mx + b form)
- Slope = 5, b = 2 -> y-int = (0, 2)
- < so dotted line and shade below
Problem 2
How do we graph a system of Inequalities?
- 2 inequalities on the same graph
Graph a System of Inequalities
Steps to Graph a System of Inequalities
1. Graph the first inequality.
2. Graph the second inequality.
3. Shade where both inequalities are true.
Steps to Graph a System
Example 1
ex 1) 3y - x < 9 and (1/3)y - 1 ≥ x
1st inequality:
Isolate y.
3y - x < 9
+ x + x
3y < x + 9
(1/3)[3y < x + 9]
y < (1/3)x + 3
- slope = (1/3), b = 3 -> y-int : (0, 3)
- < so dotted and shade below
2nd inequality:
Isolate y.
(1/3)y - 1 ≥ x
+ 1 + 1
(1/3)y ≥ x + 1
3[(1/3)y ≥ x + 1]
y ≥ 3x + 3
- slope = 3, b = 3 -> y-int :(0, 3)
- ≥ so solid and shade above
Example 1
Now Graph!
Graph
- Blue: shading from y ≥ 3x + 3
- Red: shading from y < (1/3)x + 3
- Blue and Red: where they overlap - (final answer/shading)
Graph
Now, you try!
- From our previous "Now, you try!", we obtained the graph of two inequalities.
- What happens when we graph both inequalities on the same graph? (called a system of inequalities)
- Where will the shading for the first graph be? The second graph?
- Where will the shading for the final graph be? Why?
1. y ≤ x + 1 and y < 5x + 2
Note that: These are the same problems we simplified and graphed on the previous "Now, you try!" combined into one system.
Graph of the 1st Inequality
- y ≤ x + 1
Graph of the 1st Inequality
Graph of the 2nd Inequality
- y < 5x + 2
Graph of the 2nd Inequality
Graph of the System of Inequalities
- y ≤ x + 1 and y < 5x + 2 on the same graph
- Blue and red where both graphs are shaded (solution)
Graph of the system
Special Cases
- What if there is no y to solve for? Solve for x instead!
- Line cuts through the x-axis and is vertical.
- <, ≤ shade to the left (where x is smaller)
- >, ≥ shade to the right (where x is bigger)
- All other rules stay the same.
ex) x < 5
- No y -> line is vertical and cuts through the x-axis at 5
- < so dotted and shade to the left
Special Cases
Special Cases (cont.)
- What if there is no x in your equation?
- This means there is no slope.
- Line cuts through the y-axis and is horizontal.
- All other rules stay the same.
ex) y > 2
- No x -> line cuts through the y-axis at 2 and is horizontal
- > so dotted and shade up
Special Cases cont.
Example
ex) 5y - 25 ≤ 0 and -2x + 4 ≤ 0
Example
Inequality 2:
-2x + 4 ≤ 0
No y -> Solve for x.
-2x + 4 ≤ 0
- 4 - 4
-2x ≤ -4
(-1/2)[-2x ≤ -4] (Multiply by -1 so flip inequality sign)
x ≥ 2
- No y -> vertical line that cuts through the x-axis at 2
- ≥ so solid and shade to the right
Inequality 1:
5y - 25 ≤ 0
Solve for y.
5y - 25 ≤ 0
+ 25 + 25
5y ≤ 25
(1/5)[5y ≤ 25]
y ≤ 5
- No x -> horizontal line that cuts through the y-axis at 5
- ≤ so solid and shade below
Graph
- Red and Blue where both graphs are shaded (final solution)
Graph
Example
ex) 5y - 25x ≤ 0 and -2x + 8 > 0
Example
Inequality 1:
5y - 25x ≤ 0
Solve for y.
5y - 25x ≤ 0
+ 25x + 25x
5y ≤ 25x
(1/5)[5y ≤ 25x]
y ≤ 5x
- Slope = 5, b = 0 -> y-int: (0, 0)
- ≤ so solid and shade below
Inequality 2:
-2x + 8 > 0
No y -> Solve for x.
-2x + 8 > 0
- 8 - 8
-2x > -8
(-1/2)[-2x > -8] (Multiply by -1 so flip inequality sign)
x < 4
- No y -> vertical line that cuts through the x-axis at 4
- < so dotted and shade to the left
Graph
- Red and Blue where both graphs are shaded (final solution)
Graph
Now, you try!
- What will the graph be for the following system of inequalities?
- How will you graph and shade when there is no y? When there is no x?
- Where will the final solution be shaded from both inequalities? Why?
- How does this compare to the graph of a system of equations?
1. 3y - x < 6 and 5y + x > 10 + x
Now, you try!
Solution
1. 3y - x < 6 and 5y + x > 10 + x
Solution
Inequality 1:
3y - x < 6
Isolate y.
3y - x < 6
+ x + x
3y < x + 6
(1/3)[3y < x + 6]
y < (1/3)x + 2
- Slope = (1/3), b = 2 -> Y-int : (0, 2)
- < so dotted and shade below
Inequality 2:
5y + x > 10 + x
Isolate y.
5y + x > 10 + x
- x - x
5y > 10
(1/5)[5y > 10]
y > 2
- No x -> line is horizontal and cuts through the y-axis at 2
- > so dotted and shade above
Graph
- Red and blue: where both graphs are shaded (solution set)