The document provides an interview summary of a lesson on solving various types of inequalities: 1) The lesson covers linear, quadratic, and rational inequalities, explaining the steps to solve each type. 2) Examples are worked through demonstrating how to identify intervals on the number line and use test values to determine the solution set of inequalities. 3) Applications involving break-even points and projectile motion are presented to show real-world examples.

1. 1.7 - 1Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1
Equations and
Inequalities
Interview with ....
Solving Inequalities
By
Waleed Abuelela
March 12, 2019

2. By the end of this lesson you
should know how to solve
• Linear Inequalities
• Quadratic Inequalities
• Rational Inequalities

3. What do you know about inequalities?
What are differences between them and
equations?
Do inequalities have special properties?
The goal of such questions is to:
Understand what students are thinking.
Elicit several answers from the students.

4. Properties of Inequality
Let a, b and c represent real numbers.
1. If a < b, then a ± c < b ± c.
2. If a < b and if c > 0, then ac < bc.
3. If a < b and if c < 0, then ac > bc.
4.If a < b and if c > 0, then a/c < b/c.
5. If a < b and if c < 0, then a/c > b/c.
Replacing < with >, ≤ , or ≥ results in similar properties.

5. Can we classify inequalities?
According to what?
Encourage students to jot down answers, then
combine answers in a small group.

6. Linear Inequality in One
Variable
A linear inequality in one variable is
an inequality that can be written in the
form
where a and b are real numbers, with
a ≠ 0. (Any of the symbols ≥, <, and ≤
may also be used.)
0,ax b+ >

7. As we did with linear equations,
we are looking for the solution set
of the linear inequality.
Invite students to talk about linear eq., other types of
eq’s., and refresh their memories about sets as well
as the concept of sol. set.

8. Example SOLVING A LINEAR INEQUALITY
Solve
Solution
3 5 7.x− + > −
3 5 7x− + > −
53 7 55x− > −− −+ Subtract 5.
3 12x− > −
3 12
3 3
x
<
− −
− −
Divide by –3. Reverse
the direction of the
inequality symbol
when multiplying or
dividing by a negative
number.
Don’t forget to
reverse the
symbol here.
< = −∞4 ( ,4)x
Combine like terms.
Ask them, why is this linear?

9. Here (and after any major point) one can pause and show
one or two multiple-choice questions, call for a vote (turn
this into a competition or a debate).
Solve
A.
B.
C.
D.
2
12 4
5
x
x
−
≥ −
10
3
x ≤
30
11
x ≤
10
3
x ≥
30
11
x ≥

10. Application FINDING THE BREAK-EVEN
POINT
If the revenue and cost of a certain product
are given by
4R x= and 2 1000,C x= +
where x is the number of units produced and
sold, at what production level does R at least
equal C ?
Let them try and guide them to the right thinking.

11. Implementing applications is attracting students and showing the beauty
of mathematics. Let alone its crucial role in satisfying Bloom’s taxonomy

12. Application FINDING THE BREAK-EVEN
POINT
Solution For R at least equal C , this means R ≥ C .
4 2 1000x x≥ + Substitute.
2 1000x ≥ Subtract 2x.
500x ≥ Divide by 2.
The break-even point (all costs that must be paid are paid,
and there is neither profit nor loss) is at x = 500. This
product will at least break even if the number of units
produced and sold is in the interval [500, ∞).

13. When finishing some part of the lecture, allow time for questions.
Doing so helps students to sort out the information. Then ask a
question that motivates the next part.
What if the inequality isn’t linear?
What if we are dealing with quadratic inequality?
What does quadratic inequality look like?
How can we solve it?
This kind of questions motivates students and plays a fundamental
role in brainstorming.

14. Quadratic Inequalities
A quadratic inequality is an inequality
that can be written in the form
2
0ax bx c+ + <
for real numbers a, b, and c, with a ≠
0. (The symbol < can be replaced with
>, ≤, or ≥.)

15. Solving a Quadratic Inequality
Step 1 Solve the corresponding quadratic
equation.
Step 2 Identify the intervals determined by
the solutions of the equation.
Step 3 Use a test value from each interval
to determine which intervals form
the solution set.

16. Example SOLVING A QUADRATIC
INEQULITY
Solve
Solution
2
12 0.x x− − <
Step 1 Find the values of x that satisfy
x2
– x – 12 = 0.
( 3)( 4) 0x x+ − = Factor.
3 0x + = 4 0x − =or Zero-factor property
3x = − 4x =or Solve each equation.
Ask students: How can we solve this equation?

17. Example SOLVING A QUADRATIC
INEQULITY
Step 2
–3 0 4
Interval A Interval B Interval C
(–∞, – 3) (–3, 4) (4, ∞)

18. Example SOLVING A QUADRATIC
INEQULITY
Interval
Test
Value
Is x2
–x – 12 < 0
True or False?
A: (–∞, –3) –4 (–4)2
– (–4) – 12 < 0 ?
8 < 0 False
B: (–3, 4) 0 02
– 0 – 12 < 0 ?
–12 < 0 True
C: (4, ∞) 5 52
– 5 – 12 < 0 ?
8 < 0 False
Step 3 Choose a test value from each interval.
Since the values in Interval B make the inequality
true, the solution set is (–3, 4).

19. Group work SOLVING A QUADRATIC
INEQUALITY
Solve the inequality
Solution
2
2 5 12 0.x x+ − ≥
Step 1 Find the values of x that satisfy
Corresponding
quadratic equation
(2 3)( 4) 0x x− + = Factor.
2 3 0x − = or 4 0x + = Zero-factor property
2
2 5 12 0.x x+ − =
Divide students into small groups and let them solve together. This will indeed develop the students’ ability to
communicate and work in teams. After they finish let students choose one of them to go to the board to show their work.
•

20. Group work SOLVING A QUADRATIC
INEQUALITY
Solve
Solution
2
2 5 12 0.x x+ − ≥
Step 1
2 3 0x − = or 4 0x + =
3
2
x = or 4x = −

21. Group work SOLVING A QUADRATIC
INEQUALITY
Solve
Solution
2
2 5 12 0.x x+ − ≥
Step 2 The values form the intervals on
the number line.
–4
0
3/2
Interval A Interval B Interval C
(–∞, – 4] [– 4, 3/2] [3/2, ∞)

22. Group work SOLVING A QUADRATIC
INEQUALITY
Step3 Choose a test value from each interval.
Interval
Test
Value
Is 2x2
+ 5x – 12 ≥ 0
True or False?
A: (–∞, –4] –5 2(–5)2
+5(–5) – 12 ≥ 0 ?
13 ≥ 0 True
B: [– 4, 3/2] 0 2(0)2
+5(0) – 12 ≥ 0 ?
–12 ≥ 0 False
C: [3/2, ∞) 2 2(2)2
+ 5(2) – 12 ≥ 0 ?
6 ≥ 0 True
The values in Intervals A and C make the inequality true, so
the solution set is the union of the intervals  
−∞ − ∪ ∞÷ 
3
( , 4] , .
2

23. Application FINDING PROJECTILE HEIGHT
If a projectile is launched from ground level
with an initial velocity of 96 ft per sec, its
height s (in feet) t seconds after launching is
given by the following equation.
2
16 96s t t= − +
When will the projectile be greater than 80 ft
above ground level?
Here one should invite students to give some thoughts.
Let them participate.

24. Application FINDING PROJECTILE HEIGHT
2
16 96 80t t− + >
Set s greater than
80.
Solution
2
16 96 80 0t t− + − > Subtract 80.
2
6 5 0t t + <− Divide by – 16.
Reverse the
direction of the
inequality
symbol.
Now solve the
corresponding equation.

25. Application FINDING PROJECTILE HEIGHT
2
6 5 0t t− + =
Solution
( 1)( 5) 0t t− − = Factor.
1t =
Zero-factor property
or 5t = Solve each equation.
or1 0t − = 5 0t − =

26. Application FINDING PROJECTILE HEIGHT
Solution
Interval C
1
0
5
Interval A Interval B
(–∞, 1) (1, 5) (5, ∞)
By testing the intervals, we observe that
values in Interval B, (1, 5) , satisfy the
inequality. The projectile is greater than 80
ft above ground level between 1 and 5 sec
after it is launched.

27. Solving a Rational Inequality
Step 1 Rewrite the inequality, if necessary, so that
0 is on one side and there is a single fraction on the
other side.
Step 2 Determine the values that will cause either
the numerator or the denominator of the rational
expression to equal 0. These values determine the
intervals of the number line to consider.

28. Solving a Rational Inequality
Step 3 Use a test value from each interval to
determine which intervals form the solution set.
A value causing the denominator to equal zero will
never be included in the solution set. If the
inequality is strict, any value causing the numerator
to equal zero will be excluded. If the inequality is
nonstrict, any such value will be included.

29. Can you apply these steps?

30. Try
individually
Solve
Solution
5
1.
4x
≥
+
Step 1 5
1 0
4x
− ≥
+
Subtract 1 so that 0 is
on one side.
( )5 4
0
4
x
x
− +
≥
+
Write as a single fraction.
SOLVING A RATIONAL INEQUALITY
Solving individually improves the ability to concentrate
easier and work faster and helps in getting the whole credit
for the work. (Some students prefer this)
1
4
0
x
x
−
+
≥
Combine terms in the
numerator, being careful
with signs.

31. Example
Solve
Solution
5
1.
4x
≥
+
Step 2 The quotient possibly changes sign
only where x-values make the numerator or
denominator 0. This occurs at
1 0x− = or 4 0x + =
1x = or 4x = −
SOLVING A RATIONAL INEQUALITY
Try individually

32. Example
Solve
Solution
5
1.
4x
≥
+
Step 2
–4
0
1
Interval A Interval B Interval C
(–∞, – 4) (–4, 1] [1, ∞)
SOLVING A RATIONAL INEQUALITY
Try individually

33. HOMEWORK 4
Step 3 Choose test values.
Interval
Test
Value
A: (–∞, –4) – 5
B: (–4, 1] 0
C: [1, ∞) 2
5
Is 1 True or False?
4x
≥
+
?
5
1
5 4
≥
− +
5 1 False− ≥
0
?
5
1
4
≥
+ 5
1 e
4
Tru≥
2
?
5
1
4
≥
+
5
1 e
6
Fals≥
SOLVING A RATIONAL
INEQUALITY
Try individually

34. Example 8
Step 3
The values in the interval (–4, 1) satisfy the
original inequality. The value 1 makes the
nonstrict inequality true, so it must be included
in the solution set. Since –4 makes the
denominator 0, it must be excluded. The
solution set is (–4, 1].
SOLVING A RATIONAL INEQUALITYTry
individually

35. Today’s lesson has finished
Your leaving ticket is to
sum up
Ask students to say one or two sentences about the lesson before
they leave.

36. Thanks fo r listening