Linear Inequalities - Special Cases

Table of Contents

1. Three Possible Outcomes
2. Case 1: No Solution (Empty Set)
3. Case 2: All Real Numbers
4. Case 3: Strict vs. Non-Strict at the Boundary
5. How to Recognize Special Cases
6. More Practice Examples
7. Practice Exercises
8. Related Articles

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1. Three Possible Outcomes

When solving a linear inequality, there are three possible outcomes:

Key idea: Special cases happen when the variable cancels out completely, leaving a statement with only numbers.

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2. Case 1: No Solution (Empty Set)

An inequality has no solution when simplifying leads to a false numerical statement.

Example 1

Solve: $2x+5>2x+9$

Subtract $2x$ from both sides:

This is false --- no matter what value of $x$ we choose, the left side will always be 4 less than the right side.

Solution: $x\in \varnothing$ (no solution)

Example 2

Solve: $3(x-1)\le 3x-7$

Expand:

Subtract $3x$:

This is false ($-3$ is greater than $-7$, not less).

Solution: $x\in \varnothing$

Example 3

Solve: $4x+8<4(x+1)$

Expand the right side:

Subtract $4x$:

This is false.

Solution: $x\in \varnothing$

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3. Case 2: All Real Numbers

An inequality is satisfied by all real numbers when simplifying leads to a true numerical statement.

Example 1

Solve: $2x+5<2x+9$

Subtract $2x$ from both sides:

This is always true --- no matter what $x$ equals, the left side is always 4 less than the right side.

Solution: $x\in (-\infty ,\infty )=\mathbb{R}$

Example 2

Solve: $3(x+2)\ge 3x+1$

Expand:

Subtract $3x$:

This is always true.

Solution: $x\in \mathbb{R}$

Example 3

Solve: $x-4\le x+10$

Subtract $x$:

This is always true.

Solution: $x\in \mathbb{R}$

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4. Case 3: Strict vs. Non-Strict at the Boundary

A subtle but important case occurs when the variable cancels out and leaves $0$ compared to $0$.

With Strict Inequality

Solve: $5x+3>5x+3$

Subtract $5x+3$:

This is false. Zero is not greater than zero.

Solution: $x\in \varnothing$

With Non-Strict Inequality

Solve: $5x+3\ge 5x+3$

Subtract $5x+3$:

This is true. Zero is equal to zero, and $\ge$ includes equality.

Solution: $x\in \mathbb{R}$

Notice: The only difference between these two problems is the inequality symbol ($>$ vs. $\ge$), but they have completely different solutions!

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5. How to Recognize Special Cases

You can often spot special cases before solving completely:

Pattern 1: Identical Variable Terms

If both sides have the same $x$-term, the variable will cancel out.

After subtracting $2x$: $3>7$ --- a numerical statement.

Pattern 2: Multiplied by the Same Factor

Expanding: $3x+3\le 3x+5$. Both sides have $3x$, so it will cancel.

Decision Flowchart

After the variable cancels out:

1. Is the resulting statement true? $\Rightarrow$ Solution is $\mathbb{R}$
2. Is the resulting statement false? $\Rightarrow$ Solution is $\varnothing$

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6. More Practice Examples

Example A

Solve: $-2(x+3)>-2x+1$

Expand: $-2x-6>-2x+1$

Subtract $-2x$: $-6>1$

False. Solution: $x\in \varnothing$

Example B

Solve: $\frac{x}{2}+4\le \frac{x}{2}+10$

Subtract $\frac{x}{2}$: $4\le 10$

True. Solution: $x\in \mathbb{R}$

Example C

Solve: $7x-7x+2<5$

Simplify: $2<5$

True. Solution: $x\in \mathbb{R}$

Example D

Solve: $x+x-2x\ge 1$

Simplify: $0\ge 1$

False. Solution: $x\in \varnothing$

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Practice Exercises

• Inequalities - Special Cases - Practice identifying special cases
• Inequalities - addition and subtraction - Compare with normal cases
• Inequalities - Mixed - Mixed problems including special cases

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Related Articles

• Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
• Linear Inequalities - Rules and Formulas - Quick reference for all rules
• Linear Inequalities - Variable on Both Sides - Where special cases often arise
• Linear Inequalities - Intervals and Sets - Notation for empty set and all reals