Inequalities: When Things Are Not Equal

Introduction

An inequality is a mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥. Unlike equations that have specific solutions, inequalities often have ranges of solutions, making them powerful tools for describing real-world constraints and limitations.

Inequalities help us answer questions like “What scores do I need to get an A?” or “How many items must we sell to make a profit?” They describe relationships where one quantity is greater than, less than, or within a certain range of another.

Inequality vs. Equation
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Equation: x + 3 = 7 (x must equal 4)
Inequality: x + 3 > 7 (x can be any number greater than 4)

The inequality has infinitely many solutions:
x = 5, x = 10, x = 100, etc.

Understanding Inequality Symbols

Basic Inequality Symbols

Inequality Symbol Meanings
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< : "less than"
> : "greater than"
≤ : "less than or equal to"
≥ : "greater than or equal to"
≠ : "not equal to"

Examples:
5 < 8 (5 is less than 8)
10 > 3 (10 is greater than 3)
x ≤ 7 (x is less than or equal to 7)
y ≥ -2 (y is greater than or equal to -2)
a ≠ 0 (a is not equal to 0)

Memory Tricks:
- The "mouth" opens toward the larger number
- Think of < as "L" for "Less than"
- ≤ means "less than OR equal to"
- ≥ means "greater than OR equal to"

Reading Inequalities:
3 < x < 7 reads "3 is less than x, and x is less than 7"
or "x is between 3 and 7"

Graphing Inequalities on Number Lines

Number Line Representations
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x > 3 (x is greater than 3):
←──────●────────→
      3
Open circle at 3 (3 not included)
Arrow points right (greater values)

x ≤ -1 (x is less than or equal to -1):
←──────●────────→
     -1
Closed circle at -1 (-1 is included)
Arrow points left (lesser values)

-2 < x ≤ 4 (x is between -2 and 4, including 4):
←──●────────●────→
  -2        4
Open circle at -2, closed circle at 4
Line segment between them

x ≥ 0 (x is greater than or equal to 0):
←──────●────────→
      0
Closed circle at 0 (0 is included)
Arrow points right

Circle Types:
○ Open circle: value NOT included (<, >)
● Closed circle: value IS included (≤, ≥)

Solving Linear Inequalities

Basic Inequality Solving

Solving inequalities is similar to solving equations, with one crucial difference: when you multiply or divide by a negative number, you must flip the inequality sign.

Inequality Solving Rules
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Same as equations:
- Add/subtract same number to both sides
- Multiply/divide by positive number

SPECIAL RULE:
When multiplying or dividing by a negative number,
FLIP the inequality sign!

Examples:

Addition/Subtraction:
x + 5 > 12
x + 5 - 5 > 12 - 5
x > 7

Multiplication/Division by Positive:
3x ≤ 15
3x ÷ 3 ≤ 15 ÷ 3
x ≤ 5

Multiplication/Division by Negative:
-2x < 10
-2x ÷ (-2) > 10 ÷ (-2)  ← Sign flips!
x > -5

Why flip the sign?
If -2 < -1, then multiplying by -1 gives:
2 > 1 (inequality flips)

One-Step Inequalities

One-Step Inequality Examples
═══════════════════════════

Type 1: Addition
x + 7 ≥ 12
x ≥ 5

Graph: ←──────●────────→
              5

Type 2: Subtraction
x - 4 < 9
x < 13

Graph: ←──────○────────→
             13

Type 3: Multiplication (positive)
3x > 18
x > 6

Graph: ←──────○────────→
              6

Type 4: Multiplication (negative)
-5x ≤ 20
x ≥ -4  ← Sign flipped!

Graph: ←──────●────────→
             -4

Type 5: Division (positive)
x/2 ≥ 8
x ≥ 16

Type 6: Division (negative)
x/(-3) < 6
x > -18  ← Sign flipped!

Checking Solutions:
For x > 6, test x = 7:
3(7) = 21 > 18 ✓

For x ≥ -4, test x = 0:
-5(0) = 0 ≤ 20 ✓

Two-Step Inequalities

Two-Step Inequality Process
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General form: ax + b < c
Step 1: Subtract b from both sides
Step 2: Divide by a (flip sign if a is negative)

Example 1: 2x + 3 ≤ 11
Step 1: 2x ≤ 8
Step 2: x ≤ 4

Graph: ←──────●────────→
              4

Example 2: -3x + 7 > 1
Step 1: -3x > -6
Step 2: x < 2  ← Sign flipped!

Graph: ←──────○────────→
              2

Example 3: 5 - 2x ≥ 13
Step 1: -2x ≥ 8
Step 2: x ≤ -4  ← Sign flipped!

Graph: ←──────●────────→
             -4

Example 4: (x + 1)/3 < 4
Step 1: x + 1 < 12
Step 2: x < 11

Checking: For x ≤ 4, test x = 0:
2(0) + 3 = 3 ≤ 11 ✓

Multi-Step Inequalities

Inequalities with Variables on Both Sides

Variables on Both Sides
══════════════════════

Example 1: 3x + 5 > x + 13
Step 1: Subtract x from both sides
2x + 5 > 13
Step 2: Subtract 5 from both sides
2x > 8
Step 3: Divide by 2
x > 4

Example 2: 7x - 2 ≤ 4x + 10
Step 1: Subtract 4x from both sides
3x - 2 ≤ 10
Step 2: Add 2 to both sides
3x ≤ 12
Step 3: Divide by 3
x ≤ 4

Example 3: 2x - 7 < 5x + 8
Step 1: Subtract 2x from both sides
-7 < 3x + 8
Step 2: Subtract 8 from both sides
-15 < 3x
Step 3: Divide by 3
-5 < x  or  x > -5

Strategy: Move variables to side with larger coefficient
This often avoids negative coefficients

Inequalities with Parentheses

Parentheses in Inequalities
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Example 1: 3(x - 2) ≥ 15
Step 1: Distribute
3x - 6 ≥ 15
Step 2: Add 6
3x ≥ 21
Step 3: Divide by 3
x ≥ 7

Example 2: 2(x + 1) < 4(x - 3)
Step 1: Distribute both sides
2x + 2 < 4x - 12
Step 2: Subtract 2x
2 < 2x - 12
Step 3: Add 12
14 < 2x
Step 4: Divide by 2
7 < x  or  x > 7

Example 3: -2(3x - 1) > 10
Step 1: Distribute
-6x + 2 > 10
Step 2: Subtract 2
-6x > 8
Step 3: Divide by -6 (flip sign!)
x < -4/3

Example 4: 5 - 3(x + 2) ≤ 8
Step 1: Distribute
5 - 3x - 6 ≤ 8
Step 2: Combine like terms
-1 - 3x ≤ 8
Step 3: Add 1
-3x ≤ 9
Step 4: Divide by -3 (flip sign!)
x ≥ -3

Compound Inequalities

“And” Compound Inequalities

These represent values that satisfy both conditions simultaneously.

"And" Compound Inequalities
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Form: a < x < b (x is between a and b)
This means: x > a AND x < b

Example 1: -3 < x < 5
Solution: All numbers between -3 and 5
Graph: ←──○────────○──→
         -3        5

Example 2: Solve 1 < 2x + 3 < 9
Method: Solve both parts simultaneously
1 < 2x + 3 < 9
Subtract 3 from all parts:
1 - 3 < 2x < 9 - 3
-2 < 2x < 6
Divide all parts by 2:
-1 < x < 3

Graph: ←──○────────○──→
         -1        3

Example 3: Solve -4 ≤ 3x - 1 ≤ 8
-4 ≤ 3x - 1 ≤ 8
Add 1 to all parts:
-3 ≤ 3x ≤ 9
Divide by 3:
-1 ≤ x ≤ 3

Graph: ←──●────────●──→
         -1        3

Checking: Test x = 0 (should work)
-4 ≤ 3(0) - 1 ≤ 8
-4 ≤ -1 ≤ 8 ✓

“Or” Compound Inequalities

These represent values that satisfy at least one condition.

"Or" Compound Inequalities
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Form: x < a OR x > b
Solution: Two separate regions

Example 1: x < -2 OR x > 4
Graph: ←──○────────○──→
         -2        4
       ←──          ──→

Example 2: Solve 2x + 1 < -3 OR 2x + 1 > 7
Solve each inequality separately:

Left side: 2x + 1 < -3
2x < -4
x < -2

Right side: 2x + 1 > 7
2x > 6
x > 3

Solution: x < -2 OR x > 3

Graph: ←──○────────○──→
         -2        3
       ←──          ──→

Example 3: |x| > 5
This means: x < -5 OR x > 5
(Distance from 0 is greater than 5)

Graph: ←──○────────○──→
         -5        5
       ←──          ──→

No Solution Case:
x > 5 AND x < 2
This is impossible (no overlap)
Answer: No solution or ∅

All Real Numbers Case:
x > 2 OR x < 5
This covers all real numbers
Answer: All real numbers or (-∞, ∞)

Absolute Value Inequalities

Understanding Absolute Value Inequalities

Absolute value represents distance from zero, so absolute value inequalities describe ranges of distances.

Absolute Value Inequality Types
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Type 1: |x| < a (distance less than a)
Solution: -a < x < a
Graph: ←──○────────○──→
         -a        a

Type 2: |x| > a (distance greater than a)
Solution: x < -a OR x > a
Graph: ←──○────────○──→
         -a        a
       ←──          ──→

Examples:

|x| < 3
Solution: -3 < x < 3
Graph: ←──○────────○──→
         -3        3

|x| ≥ 2
Solution: x ≤ -2 OR x ≥ 2
Graph: ←──●────────●──→
         -2        2
       ←──          ──→

|x + 1| < 4
Think: Distance from -1 is less than 4
-4 < x + 1 < 4
-5 < x < 3

|2x - 3| > 5
This means: 2x - 3 < -5 OR 2x - 3 > 5

Left: 2x - 3 < -5    Right: 2x - 3 > 5
      2x < -2              2x > 8
      x < -1               x > 4

Solution: x < -1 OR x > 4

Word Problems with Inequalities

Translating Word Problems

Inequality Word Problem Strategy
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Key Phrases:
"at least" → ≥
"at most" → ≤
"more than" → >
"less than" → <
"between" → compound inequality
"maximum" → ≤
"minimum" → ≥

Example 1: Grade Requirements
"To get an A, you need at least 90% average on 4 tests.
Your first three scores are 85, 92, and 88. What score
do you need on the fourth test?"

Let x = fourth test score
Average ≥ 90
(85 + 92 + 88 + x)/4 ≥ 90
(265 + x)/4 ≥ 90
265 + x ≥ 360
x ≥ 95

You need at least 95% on the fourth test.

Example 2: Budget Constraints
"You have $50 to spend on books. Each book costs $12.
How many books can you buy?"

Let x = number of books
Cost ≤ Budget
12x ≤ 50
x ≤ 4.17...

Since you can't buy part of a book: x ≤ 4
You can buy at most 4 books.

Example 3: Temperature Range
"The temperature must be between 68°F and 72°F.
Write an inequality for Celsius temperature."

F = (9/5)C + 32
68 ≤ (9/5)C + 32 ≤ 72
36 ≤ (9/5)C ≤ 40
20 ≤ C ≤ 22.22...

The Celsius temperature must be between 20°C and 22.2°C.

Business and Economics Applications

Real-World Inequality Applications
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Profit Analysis:
"A company makes $15 profit per item but has $300 in
fixed costs. How many items must they sell to make
at least $500 profit?"

Let x = number of items
Profit = Revenue - Costs
15x - 300 ≥ 500
15x ≥ 800
x ≥ 53.33...

They must sell at least 54 items.

Break-Even Analysis:
"Revenue is $25 per item, costs are $18 per item plus
$140 fixed costs. How many items for break-even?"

Revenue ≥ Costs
25x ≥ 18x + 140
7x ≥ 140
x ≥ 20

Need to sell at least 20 items to break even.

Shipping Constraints:
"A box can hold at most 50 pounds. Small items weigh
2 pounds each, large items weigh 5 pounds each. If you
have 8 small items, how many large items can you add?"

Let x = number of large items
Total weight ≤ 50
2(8) + 5x ≤ 50
16 + 5x ≤ 50
5x ≤ 34
x ≤ 6.8

You can add at most 6 large items.

Investment Planning:
"You want to invest in stocks (risky) and bonds (safe).
You have $10,000 and want at most 30% in stocks.
How much can you invest in stocks?"

Let x = amount in stocks
x ≤ 0.30(10,000)
x ≤ 3,000

You can invest at most $3,000 in stocks.

Graphing Inequalities

Coordinate Plane Inequalities

Graphing Linear Inequalities
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Steps:
1. Graph the boundary line (y = mx + b)
2. Use solid line for ≤ or ≥
3. Use dashed line for < or >
4. Shade appropriate region
5. Test a point to verify

Example 1: y > 2x + 1
Step 1: Graph y = 2x + 1 (dashed line)
Step 2: Shade above the line (y > means above)

Test point (0, 0):
0 > 2(0) + 1
0 > 1 (False)
So (0,0) is NOT in solution region
Shade the region that doesn't contain (0,0)

Example 2: y ≤ -x + 3
Step 1: Graph y = -x + 3 (solid line)
Step 2: Shade below the line (y ≤ means below)

Test point (0, 0):
0 ≤ -(0) + 3
0 ≤ 3 (True)
So (0,0) IS in solution region
Shade the region containing (0,0)

Example 3: x ≥ -2
This is a vertical line at x = -2
Shade to the right (x ≥ means right of line)

Memory Aid:
y > or y ≥ → shade ABOVE
y < or y ≤ → shade BELOW
x > or x ≥ → shade RIGHT
x < or x ≤ → shade LEFT

Common Mistakes and Solutions

Typical Inequality Errors

Common Inequality Mistakes
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Mistake 1: Not Flipping Sign with Negatives
Wrong: -2x < 6, so x < -3
Right: -2x < 6, so x > -3

Solution: Always flip when multiplying/dividing by negative

Mistake 2: Wrong Circle Type on Graph
Wrong: x > 3 with closed circle ●
Right: x > 3 with open circle ○

Solution:
○ for < and > (not included)
● for ≤ and ≥ (included)

Mistake 3: Compound Inequality Errors
Wrong: x < 2 OR x > 5 written as 2 > x > 5
Right: x < 2 OR x > 5 (two separate regions)

Solution: "AND" gives overlap, "OR" gives union

Mistake 4: Absolute Value Confusion
Wrong: |x| > 3 means -3 > x > 3
Right: |x| > 3 means x < -3 OR x > 3

Solution:
|x| < a → -a < x < a (between)
|x| > a → x < -a OR x > a (outside)

Mistake 5: Word Problem Translation
Wrong: "at most 5" means x > 5
Right: "at most 5" means x ≤ 5

Solution: Learn key phrase meanings

Conclusion

Inequalities extend our problem-solving toolkit beyond exact solutions to ranges and constraints. They model real-world situations where we need to find acceptable ranges rather than precise values.

Inequalities: Complete Understanding
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Conceptual Understanding:
✓ Inequalities as comparisons between expressions
✓ Solution sets as ranges rather than specific values
✓ Absolute value as distance from zero

Procedural Fluency:
✓ Solving linear inequalities systematically
✓ Handling compound and absolute value inequalities
✓ Graphing solutions on number lines and coordinate planes

Strategic Competence:
✓ Translating word problems involving constraints
✓ Choosing appropriate inequality symbols
✓ Interpreting solutions in context

Adaptive Reasoning:
✓ Understanding why signs flip with negative multiplication
✓ Recognizing when to use "and" vs "or" logic
✓ Making connections between algebraic and graphical representations

Productive Disposition:
✓ Confidence working with ranges and constraints
✓ Appreciation for inequality applications in real life
✓ Persistence in multi-step inequality problems

From quality control in manufacturing to financial planning, from sports statistics to scientific research, inequalities provide essential tools for describing and working with constraints, limitations, and acceptable ranges in countless real-world applications.