MHF4U
Ms. Copija Heritage Academy
Course: MHF4U – Advanced Functions, Grade 12, University Preparation
Unit
Polynomial Functions
Activity/Lesson Title
Lesson #7 - Inequalities
Strands
Polynomial and Rational Functions
 Solving Inequalities
Expectations:
Overall Expectations:
 C4 demonstrate an understanding of solving polynomial and
simple rational inequalities
Specific Expectations:
 C4.1 explain, for polynomial and simple rational functions,
the difference between the solution to an equation in one
variable and the solution to an inequality in one variable, and
demonstrate that given solutions satisfy an inequality (e.g.,
demonstrate numerically and graphically that the solution to
< 5 is x < –1 or x > – );
Planning Notes:

C4.2 determine solutions to polynomial inequalities in one
variable [e.g., solve f(x) ≥ 0, where f(x) = x – x + 3x – 9]
and to simple rational inequalities in one variable by
graphing the corresponding functions, using graphing
technology, and identifying intervals for which x satisfies the
inequalities

C4.3 solve linear inequalities and factorable polynomial
inequalities in one variable (e.g., x + x > 0) in a variety of
ways (e.g., by determining intervals using x-intercepts and
evaluating the corresponding function for a single x-value
within each interval; by factoring the polynomial and
identifying the conditions for which the product satisfies the
inequality), and represent the solutions on a number line or
algebraically (e.g., for the inequality x – 5x + 4 < 0, the
solution represented algebraically is – 2 < x < –1 or 1 < x <
2)

Make student copies of
o Solve Inequalities by Graphing
o Solving Inequalities Algebraically
Teaching and Learning Anticipatory Set
Strategies
 Teacher Directed Review: Work through solving
Linear Inequalities. Remind students how to
solve linear inequalities.
10-15
mins
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Culminating Activity
Ms. Copija Heritage Academy
Activity
 Whole Class Investigation: Work through
“Solving Inequalities by Graphing” as a class.
Read information with students, and together fill
in the blanks on the handouts.
 Drill and Practice: Students will solve two
inequalities by graphing as practice.
 Whole Class Investigation: Work through
“Solving Inequalities Algebraically” with
students. Read through information and together
fill in the charts. Solve by factoring in method 2.
 Drill and Practice: Students will complete
practice questions from the textbook.
60 mins
Closing/Discussion
 Teacher Directed Instruction: Together as a class
we will summarize the findings of this lesson into
a key ideas summary note.
5 mins


Assessment/
Evaluative Techniques


Students will complete two investigations to help understand
the characteristics of polynomial functions
Students will complete some extra practice questions
Formative: Observational notes – teacher will observe
individual work on the practice questions to determine level
of understanding
Summative: Students will be evaluated on their
understanding through a unit test.
Homework

Complete practice questions
Accommodations/
Modifications


Printing worksheets on coloured paper with 14 font
Providing a variety of learning tools, such as calculators for
completing numeracy tasks
Providing extra basic practice questions for additional
practice
Providing problems to solve that extend learning
Simplifying the language of instruction
Providing opportunities of performance in areas of special
talent
Giving students extra time to complete classroom
assignments
Allowing students to complete tasks or present information
in alternative ways (i.e. verbal explanation)
Providing all students with strategies for understanding and
accepting exceptional students and integrating them into the
regular classroom
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





MHF4U
Resources
Ms. Copija Heritage Academy
Kirkpatrick, C., McLeish, R, Montesanto, R., Suurtamm, C., Trew, S.,
Zimmer, D., Advanced Functions and Introductory Calculus,
Nelson., Canada, 2002. ISBN: 0-17-615778-6
Ontario Association for Mathematics Education, OAME Support
Resources for MHF4U (Unit 1 Polynomial Functions),
http://www.oame.on.ca/main/index1.php?code=grspecres&ph
=12&sp=MHF4U
MHF4U
Ms. Copija Heritage Academy
Inequalities – Solving Graphically
Solving linear inequalities, such as "x + 3 > 0", is pretty straightforward, as long as
you remember to flip the inequality sign whenever you multiply or divide through
by a negative (as you would when solving something like "–2x < 4").
Linear Example:
Solve x – 4 < 0
You already know that, to solve this inequality, all you have to do is add the 4 to
the other side to get the solution "x < 4". So we already know what the answer
is. But let's approach this problem from a different angle, by considering the
related two-variable graph.
For "x – 4 < 0", the associated two-variable linear graph is y = x – 4. Sketch the
graph.
y
The inequality "x – 4 < 0" is asking "when is the
line y = x – 4 below the line y = 0?"
Since the line y = 0 is just the x-axis, the
inequality is therefore asking "when is the line
y = x – 4 below the x-axis?"
The first step in answering this question is to find
where the line crosses the x-axis;
x intercept is _______________
6
4
2
-1
1
2
3
4
-2
-4
-6
The original question asked us to solve x – 4 < 0, so we need to find where the line
is below the x-axis. This happens on the left-hand side of the intercept.
Since it is the domain we are interested in, we can restrict our graph to the x-axis (a
number line)
5
6
x
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Ms. Copija Heritage Academy
Quadratic Example #1:
ca
Solve -x2 + 4 < 0.
First, we need to look at the associated two-variable equation, y = -x2 + 4, and
consider where its graph is below the x-axis. To do this, I need to know where the
graph crosses the x-axis.
1
5
4
3
2
0– 2
1
2
3
4
5
3
4
5
1
The zeros are at x = _____ and at x = _____.
This divides the number line into three intervals:
–5 –4 –3 –2 –1 0
– 5 – 4 – 3 – 2 – 2– 1 0
1
1
2
2
3
4
5
2 3 4 5
Now we need to figure out where (on which intervals) the graph is below the axis.
Since this is a "negative" quadratic, it opens __________.
y
6
To solve the original inequality, I need to
find the intervals where the graph is below
the axis (so the y-values are less than
zero). Sketch the graph of y = –x2 + 4
4
2
-6
-4
-2
2
-2
-4
-6
Our knowledge of graphing, together with the zeroes, tells us that we want the
intervals on either end, rather than the interval in the middle.
Then the solution is clearly: x < –2 and x > 2
4
6
x
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Ms. Copija Heritage Academy
Quadratic Example #2:
ca
2
Solve x  3x  15  5
y
x





Cubic Example:
ca
3
2
Solve x  4 x  x  6  0






y
x











MHF4U
Ms. Copija Heritage Academy
Inequalities – Solving Algebraically
In the intervals between the x-intercepts, the function is either positive or negative.
There are two different algebraic ways of checking for “positivity” or “negativity”
over given intervals.
1) Test-point method
Pick a point (any point) in each interval. Calculate the value of y at that point.
Whatever the sign on that value is, that is the sign for that interval.
Solve x 2  3x  2  0
i)
set x 2  3x  2  0 and solve for the intercepts


ii)
iii)
iv)
v)
complete the intervals in the table
select a point in each of the intervals
calculate the value of y at each point
record the sign of the y value
Interval
test value for x
y value
Sign of the y value
x<
<x<
<x
The sign of the y value is the sign of the entire interval. Decide which intervals
satisfy the inequality.
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Ms. Copija Heritage Academy
2) Factor method
Calculate the signs of the factors and the product of their signs
Solve x 2  3x  2  0
First factor y = x2 – 3x + 2

Set up a grid showing the factors and the intervals.
Sign of x  1
Sign of x  2
Sign of ( x  1)( x  2)
x1
1 x  2
2x
 The factor x – 1 is positive for x > 1 so we place a plus sign when x is
between 1 and 2. Similarly x – 1 is positive for 2  x so we also place a
plus sign in that interval.
 x – 2 is positive for x > 2, so we place a plus sign in this interval.
 Place a minus sign in any region that isn’t positive.
 Now multiply down the columns to compute the sign of y on each
interval.
recall:
(plus)×(plus) = (plus)
(minus)×(minus) = (plus)
(minus)×(plus) = (minus)
State the intervals where y > 0
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Cubic Example:
Solve x 3  4 x 2  x  6  0 using any method.
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MHF4U
Ms. Copija Heritage Academy
Inequalities – Solving Graphically (Practice)
1. Solve the following by graphing and stating the intervals.
a. 5( x  1)( x  1)  0
y
x





















b. (2 x  1)( x  3)( x  4)  0
y
x






c. (2 x  3)( x  4)( x  5)( x  2)  0
y
x






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Ms. Copija Heritage Academy
2. Let f ( x)  5( x  2)( x  2)( x  3) . Graph f ( x) . Shade the part of the
graph that shows f ( x)  0 . What interval does the shaded part represent?
y
x











3. Let f ( x)  2( x  3)( x  3)( x  1)( x  1) . Graph f ( x) . Shade the part of
the graph that shows f ( x)  0 . What interval does the shaded part
represent?
y
x











4. Communication. Create a degree-4 polynomial function, f ( x) , with three
zeros and a negative leading coefficient. Graph f ( x) and explain how to use
the zeros to solve f ( x)  0 .
y
x











MHF4U
Ms. Copija Heritage Academy
5. A computer software company models the profits on its latest game
using P(n)  2n 2  28n  90 , where n represents the number of games sold
in hundred thousands and P represents the profit in millions of dollars.
a. How many games must the company sell to break even?
b. When will the company make a profit? Lose money?
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Ms. Copija Heritage Academy
Answers:
1. a)  1  x  1
1
or 3  x
2
3
c) x  5 or  4  x  or 2  x
2
b)  4  x  
2.  3  x  2 or 2  x
3.  3  x  1 or 1  x  3
4. Show your answer to a teacher
5. a) The company must sell 500 000 or 900 000 games to break even (when
profit = 0)
b) The company will make profit when 500000  n  90000
The company will lose profit when n  500000 or when 900000  n
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Ms. Copija Heritage Academy
Inequalities – Solving Algebraically (Practice)
1. Solve the following algebraically.
a. 5( x  1)( x  1)  0
b. (2 x  1)( x  3)( x  4)  0
c. (2 x  3)( x  4)( x  5)( x  2)  0
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2. Solve the following by factoring and then finding the intervals.
a. x 2  6x  9  16
b. x 3  9 x  0
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c. x 3  2 x 2  5x  6  0
d. x 4  5x 2  4  0
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Ms. Copija Heritage Academy
3. Consider the inequality ( x  3) 2 ( x  10) 2  0 . When will the inequality be
negative? Solve algebraically.
Answers:
1. a)  1  x  1
1
or 3  x
2
3
c) x  5 or  4  x  or 2  x
2
b)  4  x  
2. a) x  1 or 7  x
b)  3  x  0 or 3  x
c)  2  x  1 or 3  x
d)  2  x  1 or 1  x  2
3. This graph will never be negative because its minimum values are when x=10 and x=3.
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Name: _________________________
Inequalities - Quiz
Solve the following algebraically.
x 3  4 x 2  7 x  10  0
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Date:___________________