Lesson 7: Graphing Inequalities

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 -

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

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

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

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

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 =

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, ∞ =

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)

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) ∅

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]

Lesson 7: Graphing Inequalities

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