Lehman PTS-3 Summer 2020

1. 𝐏𝐓𝐒 𝟑
Bridge to Calculus Workshop
Summer 2020
Lesson 7
Graphing Inequalities
“There are three types of people in
the world: those who can count, and
those who cannot." – Anonymous -

2. Lehman College, Department of Mathematics
Factoring by Grouping (1 of 1)
Example 0. Solve the quadratic equation 6𝑥2
+ 𝑥 − 15:
Solution. The factoring number is:
Step 1. Determine the two factors:
Step 2. Use the coefficients found in Step 1 above to
split the linear term in the given trinomial:
Step 2. Factor the result using grouping:
Check your answer by distributing the two factors.
6𝑥2 + 𝑥 − 15 = 6𝑥2 − 9𝑥 + 10𝑥 − 15
6𝑥2
+ 𝑥 − 15 = 6𝑥2
− 9𝑥 + 10𝑥 − 15
= 3𝑥 2𝑥 − 3 + 5 2𝑥 − 3
= 3𝑥 + 5 2𝑥 − 3
6 −15 = −90
10 and −9

3. Lehman College, Department of Mathematics
Sets and Set Notation (1 of 3)
A set is a collection of objects. These objects are called
the elements of the set. If 𝑆 is a set, and 𝑎 is an element
of 𝑆, then we write 𝑎 ∈ 𝑆. If 𝑏 is not an element of 𝑆, we
write 𝑏 ∉ 𝑆.
For example, ℤ represents the set of integers. We write
− 2 ∈ ℤ, but 𝜋 ∉ ℤ.
Some sets can be described by listing their elements
within braces. For instance, the set 𝐴 that consists of all
positive integers less than 7 can be written as:
We can also write 𝐴 in set-builder notation as:
𝑨 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕
𝑨 = 𝒙 | 𝒙 ∈ ℤ, 𝐚𝐧𝐝 𝟎 < 𝒙 < 𝟕

4. Lehman College, Department of Mathematics
Sets and Set Notation (2 of 3)
We can also write 𝐴 in set-builder notation as:
We read this as “𝐴 is the set of all 𝑥, such that 𝑥 is an
integer and 0 < 𝑥 < 7”.
If 𝑆 and 𝑇 are sets, then their union 𝑆 ∪ 𝑇 is the set that
consists of all elements that are in 𝑆 or 𝑇 (or in both).
The intersection of 𝑆 and 𝑇 is the set 𝑆 ∩ 𝑇 consisting of
all elements that are in both 𝑆 and 𝑇. That is, 𝑆 ∩ 𝑇 is
the set of elements common to both 𝑆 and 𝑇.
The empty set, denoted by ∅ is the set that contains no
element.
𝑨 = 𝒙 | 𝒙 ∈ ℤ, 𝐚𝐧𝐝 𝟎 < 𝒙 < 𝟕

5. Lehman College, Department of Mathematics
Sets and Set Notation (3 of 3)
Example 1. Suppose 𝑆 = 1, 2, 3, 4, 5 , 𝑇 = 4, 5, 6, 7 ,
and 𝑉 = 6, 7, 8 , find the sets 𝑆 ∪ 𝑇, 𝑆 ∩ 𝑇, and 𝑆 ∩ 𝑉.
Solution.
Certain sets have special notation:
𝑆 ∪ 𝑇 = 1, 2, 3, 4, 5, 6, 7 All elements in 𝑆 or 𝑇
𝑆 ∩ 𝑇 = 4, 5 Elements common to both 𝑆 and 𝑇
𝑆 ∩ 𝑉 = ∅ 𝑆 and 𝑉 have no elements common
ℕ – The set of natural (counting) numbers
ℤ – The set of integers
ℚ – The set of rational numbers
ℝ – The set of real numbers
ℂ – The set of complex numbers

6. Lehman College, Department of Mathematics
Intervals (1 of 6)
Certain sets of real numbers, called intervals, occur in
calculus and correspond to line segments. For example,
if 𝑎 < 𝑏, then the open interval from 𝑎 to 𝑏 consists of all
real numbers between 𝑎 and 𝑏 and is denoted by:
Using set-builder notation, we can write:
Note that the endpoints, 𝑎 and 𝑏, are excluded from the
interval. This fact is indicated by the parentheses and
the open circles in the graph of the interval:
(𝒂, 𝒃)
𝒙 | 𝒂 < 𝒙 < 𝒃𝒂, 𝒃 =

7. Lehman College, Department of Mathematics
Intervals (2 of 6)
If 𝑎 < 𝑏, then the closed interval from 𝑎 to 𝑏 is the set:
Here the endpoints, 𝑎 and 𝑏, are included in the
interval. This is indicated by the square brackets and
the closed circles in the graph of the interval:
It is also possible to include only one endpoint in an
interval. We also need to consider infinite-length
intervals, such as:
This does not mean that ∞ (infinity) is a real number. It
means the set extends infinitely in the positive direction.
𝒙 | 𝒂 ≤ 𝒙 ≤ 𝒃𝒂, 𝒃 =
𝒙 | 𝒂 < 𝒙𝒂, ∞ =

8. Lehman College, Department of Mathematics
Intervals (3 of 6)
The following table lists the nine possible types of
intervals. Here 𝑎 < 𝑏.
Notation Set Description Graph
𝑎, 𝑏 𝑥 | 𝑎 < 𝑥 < 𝑏
[𝑎, 𝑏] 𝑥 | 𝑎 ≤ 𝑥 ≤ 𝑏
[𝑎, 𝑏) 𝑥 | 𝑎 ≤ 𝑥 < 𝑏
(𝑎, 𝑏] 𝑥 | 𝑎 < 𝑥 ≤ 𝑏
(𝑎, ∞) 𝑥 | 𝑎 < 𝑥
[𝑎, ∞) 𝑥 | 𝑎 ≤ 𝑥
(−∞, 𝑏) 𝑥 | 𝑥 < 𝑏
(−∞, 𝑏] 𝑥 | 𝑥 ≤ 𝑏
(−∞, ∞) ℝ

9. Lehman College, Department of Mathematics
Intervals (4 of 6)
Example 2. Express each interval in terms of
inequalities, and then graph the interval:
Solution.
𝑥 | − 1 ≤ 𝑥 < 2
[−1, 2)(a) (−3, ∞)(b)
[−1, 2) =(a)
𝑥 | − 3 < 𝑥−3, ∞ =(b)

10. Lehman College, Department of Mathematics
Intervals (5 of 6)
Example 3. Graph the sets:
Solution.
𝑥 | − 2 < 𝑥 < 0 or 5 ≤ 𝑥
(−2, 0] ∪ [5, ∞)
(−2, 0] ∪ [5, ∞) =
(−2, 0] [5, ∞)
(a) −2, 0 ∩ [5, ∞)(b)
(a)
−2, 0 ∩) =(b) ∅

11. Lehman College, Department of Mathematics
Intervals (6 of 6)
Example 2. Graph each set:
Solution.
𝑥 | 1 < 𝑥 < 3 and 2 ≤ 𝑥 < 7
1, 3 ∩ [2, 7](a) 1, 3 ∪ [2, 7](b)
(a)
(b)
1, 3 ∩ 2, 7 =
𝑥 | 1 < 𝑥 < 3 or 2 ≤ 𝑥 < 71, 3 ∪ 2, 7 =
= 𝑥 | 2 ≤ 𝑥 < 3 = [2, 3)
= 𝑥 | 1 < 𝑥 ≤ 7 = (1, 7]