Algebra Mastery
The Human Knowledge Project
Chapter 6 — Inequalities
6.1 Learning Objectives
By the end of this chapter, you should be able to:
understand what inequalities represent
distinguish inequalities from equations
solve one-step inequalities
solve multi-step inequalities
graph inequalities on a number line
understand interval behavior
correctly reverse inequality signs when multiplying or dividing by negatives
solve compound inequalities
model real-world constraints using inequalities
develop structural understanding of inequality systems
6.2 Big Picture — Inequalities Describe Limits and Constraints
Equations describe exact equality.
Example:
x = 5
Inequalities describe ranges, limits, and constraints.
Example:
x > 5
This means:
x can be ANY value greater than 5
Inequalities are extremely important because real systems often involve:
limits
boundaries
tolerances
safety margins
capacities
restrictions
Examples:
maximum load limits
minimum wages
speed limits
engineering tolerances
temperature ranges
memory capacity
financial constraints
Much of reality operates through inequalities rather than exact equalities.
6.3 Inequality Symbols
Symbol Meaning
greater than
< less than
≥ greater than or equal to
≤ less than or equal to
Examples:
x > 3
means:
x is greater than 3
x ≤ 8
means:
x is less than or equal to 8
6.4 Inequalities Represent Sets of Values
Unlike equations, inequalities usually have MANY solutions.
Example:
x > 2
Possible solutions include:
3
4
100
1000
and infinitely many more.
Inequalities describe regions of possibility.
6.5 Visualizing Inequalities
Number lines help visualize inequalities.
Example:
x > 3
Graph:
open circle at 3
shading to the right
Example:
x ≥ 3
Graph:
closed circle at 3
shading to the right
Closed circles mean:
included
Open circles mean:
excluded
6.6 Solving One-Step Inequalities
Example:
x + 4 > 9
Subtract 4 from BOTH sides:
x > 5
The balance principle still applies.
6.7 Solving Subtraction Inequalities
Example:
x - 3 < 8
Add 3 to BOTH sides:
x < 11
6.8 Solving Multiplication Inequalities
Example:
3x > 12
Divide BOTH sides by 3:
x > 4
6.9 The Important Rule About Negatives
When multiplying or dividing an inequality by a NEGATIVE number:
REVERSE the inequality sign
This is one of the most important ideas in inequality solving.
6.10 Why the Sign Reverses
Example:
We know:
3 > 1
Multiply BOTH sides by:
-1
Result:
-3 < -1
The relationship reverses because negative multiplication flips direction on the number line.
6.11 Solving Negative Inequalities
Example:
-2x > 8
Divide by -2:
x < -4
Notice:
sign reverses
6.12 Multi-Step Inequalities
Example:
2x + 3 ≤ 11
Subtract 3:
2x ≤ 8
Divide by 2:
x ≤ 4
6.13 Distribution in Inequalities
Example:
3(x + 2) > 15
Distribute:
3x + 6 > 15
Subtract 6:
3x > 9
Divide by 3:
x > 3
6.14 Compound Inequalities
Compound inequalities combine multiple conditions.
Example:
2 < x < 7
This means:
x is greater than 2
AND less than 7
Possible solutions:
3
4
5
6
6.15 AND vs OR Inequalities
AND inequalities
Require BOTH conditions to be true.
Example:
1 < x < 5
OR inequalities
Require either condition to be true.
Example:
x < -2 OR x > 4
6.16 Interval Notation (Intro)
Intervals compactly describe solution sets.
Examples:
Inequality Interval
x > 3 (3, ∞)
x ≥ 3 [3, ∞)
x < 5 (-∞, 5)
2 < x < 7 (2, 7)
Parentheses mean:
excluded
Brackets mean:
included
6.17 Real-World Inequalities
Examples:
speed ≤ 65 mph
temperature > 0°C
memory usage < maximum capacity
budget ≤ available funds
load weight ≤ structural limit
Inequalities are essential for modeling safe operating ranges.
6.18 Common Beginner Difficulties
Students often struggle with:
forgetting to reverse inequality signs
graphing direction incorrectly
confusion about open vs closed circles
compound inequalities
negative arithmetic
These struggles are normal.
Inequality fluency develops through:
repetition
visualization
careful structure
troubleshooting
6.19 Mental Model
Equations identify:
exact values
Inequalities identify:
regions of possibility
Inequalities are about:
ranges
constraints
boundaries
allowable behavior
6.20 Warm-Up Problems
Problems
Solve:
x + 3 > 8
Solve:
x - 4 < 2
Solve:
2x > 10
Solve:
5x ≤ 20
Solve:
x/3 ≥ 4
Solve:
x/2 < 7
Solve:
-2x > 6
Solve:
-5x ≤ 15
Solve:
3x + 1 > 10
Solve:
2x - 5 ≤ 9
Solve:
4x + 2 < 18
Solve:
6x - 3 ≥ 9
6.21 Guided Problems
Problems
Solve:
3(x + 2) > 12
Solve:
2(x - 4) ≤ 10
Solve:
5x + 3 > 2x + 15
Solve:
7x - 4 ≤ 3x + 20
Solve:
-3x + 2 > 11
Solve:
-2(x - 5) < 8
Graph:
x > 4
Graph:
x ≤ -2
Write an inequality for:
a number greater than 7
Write an inequality for:
a number at most 12
Write an inequality for:
a speed limit of 55 mph
Explain why inequality signs reverse with negative division.
6.22 Challenge Problems
Solve:
4(x + 1) - 3 > 13
Solve:
2(3x - 4) ≤ 10
Solve:
-5x + 7 > 22
Solve:
3 - 2x ≤ 11
Solve the compound inequality:
2 < x + 1 < 8
Solve the compound inequality:
-3 ≤ 2x + 1 < 7
Explain the difference between:
equations
inequalities
Explain the meaning of:
open circles
closed circles
Describe a real-world system involving inequalities.
Explain why inequalities are important in engineering and science.
6.23 Solutions
Solutions to Warm-Up Problems
x > 5
x < 6
x > 5
x ≤ 4
x ≥ 12
x < 14
x < -3
x ≥ -3
x > 3
x ≤ 7
x < 4
x ≥ 2
Solutions to Guided Problems
x > 2
x ≤ 9
x > 4
x ≤ 6
x < -3
x > 1
Open circle at 4, shading right.
Closed circle at -2, shading left.
x > 7
x ≤ 12
s ≤ 55
Negative multiplication reverses order on the number line.
Solutions to Challenge Problems
x > 3
x ≤ 3
x < -3
x ≥ -4
1 < x < 7
-2 ≤ x < 3
Equations identify exact equality.
Inequalities identify ranges and constraints.
Open circles exclude endpoints.
Closed circles include endpoints.
Examples include:
speed limits
engineering tolerances
financial budgets
temperature ranges
memory capacity
Inequalities model safe operating limits, tolerances, capacities, and allowable ranges in real systems.