Section 2.5
Linear Inequalities in One Variable
Possible Solutions to a Linear Inequality in One Variable
Solution
Interval Notation
𝑥>2
2, ∞
𝑥≥2
2, ∞
𝑥<2
−∞, 2
𝑥≤2
−∞, 2
−1 < 𝑥 ≤ 2
−1, 2
All Real Numbers
−∞, ∞
No Solution
Graph
Big Things to Remember!!!

If you multiply or divide both sides by a
negative number, FLIP the inequality
sign.

DOUBLE CHECK to make sure you
appropriately used parentheses and
brackets.
Example 1
Solve the inequality , giving solution sets in
both interval and graph forms.
2
− 𝑥 − 6 ≤ −2
3
Example 2
Solve the inequality , giving solution sets in
both interval and graph forms.
3𝑧 − 2
−
<6
5
Example 3
Solve the inequality , giving solution sets in
both interval and graph forms.
𝑚 − 2 𝑚 − 4 ≤ 3𝑚
Example 4
Solve the inequality , giving solution sets in
both interval and graph forms.
3
1
𝑘 − 2 − 2𝑘 − 7 ≤ 3
5
4
Example 5
Solve the inequality , giving solution sets in
both interval and graph forms.
4 < −2𝑥 + 3 ≤ 8
Section 2.6
Set Operations and Compound Inequalities
Sets Operations

A set is a collection of objects called
elements of the set.

The empty set - ∅

Sets Operations
Intersection of Sets
𝐴 ∩ 𝐵 = 𝑥 𝑥 is an element of 𝐴 𝐚𝐧𝐝 𝑥 is an element of 𝐵
Union of Sets
𝐴 ∪ 𝐵 = 𝑥 𝑥 is an element of 𝐴 𝐨𝐫 𝑥 is an element of 𝐵
Example 1
Let 𝐴 = {1,2,3,4,5,6} and 𝐵 = {1,3,5}.
a.
𝐴∩𝐵 =
b.
𝐴∪𝐵 =
c.
𝐴∩∅=
d.
∅∩𝐵 =
Compound Inequalities

Example: 𝑥 < −2 or 𝑥 ≥ 3

When graphing, think . . .
◦ ∪ = or = all parts
◦ ∩ = and = overlapping parts
Example 2
Graph and give the solution set in interval
form.
a. 𝑥 < 4 and 𝑥 ≥ −3
b.
𝑥 < −3 and 𝑥 ≥ 4
Example 2 (cont.)
c.
𝑥 < −3 or 𝑥 ≥ 4
d.
𝑥 < 4 or 𝑥 ≥ −3
Example 3
Solve the inequality and give the solution
set in both interval and graph forms.
a. 7𝑥 + 6 ≤ 48 and − 4𝑥 ≥ −32
b.
3𝑥 < 𝑥 + 12 or 𝑥 + 1 > 10
Example 4
Express the set in the simplest interval
form.
−9,1 ∪ −∞, −3
Questions???

Don’t forget to check on your due dates!