Linear Inequalities - Simple Inequalities

Table of Contents

1. Type 1: Addition and Subtraction
2. Type 2: Multiplication by a Positive Number
3. Type 3: Division by a Positive Number
4. Type 4: Multiplication by a Negative Number
5. Type 5: Two Operations Combined
6. Summary
7. Practice Exercises
8. Related Articles

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Type 1: Addition and Subtraction

These are the simplest inequalities. You solve them by adding or subtracting the same number from both sides. The inequality direction does not change.

Example 1a

Solve: $x+7>12$

Check: Let $x=6$: $6+7=13>12$ $✓$ Solution: $x\in (5,\infty )$

Example 1b

Solve: $x-3\le 10$

Check: Let $x=10$: $10-3=7\le 10$ $✓$ Solution: $x\in (-\infty ,13]$

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Type 2: Multiplication by a Positive Number

When $x$ is multiplied by a positive number, divide both sides by that number. The inequality direction does not change.

Example 2a

Solve: $3x>18$

Check: Let $x=7$: $3\cdot 7=21>18$ $✓$ Solution: $x\in (6,\infty )$

Example 2b

Solve: $5x\le 40$

Check: Let $x=8$: $5\cdot 8=40\le 40$ $✓$ Solution: $x\in (-\infty ,8]$

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Type 3: Division by a Positive Number

When $x$ is divided by a positive number, multiply both sides by that number. The inequality direction does not change.

Example 3a

Solve: $\frac{x}{4}<3$

Check: Let $x=8$: $\frac{8}{4}=2<3$ $✓$ Solution: $x\in (-\infty ,12)$

Example 3b

Solve: $\frac{x}{2}\ge -5$

Check: Let $x=0$: $\frac{0}{2}=0\ge -5$ $✓$ Solution: $x\in [-10,\infty )$

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Type 4: Multiplication by a Negative Number

This is where students make the most mistakes! When you divide by a negative number, you must reverse the inequality sign.

Example 4a

Solve: $-2x>8$

Divide both sides by $-2$. Since $-2$ is negative, flip the sign:

Check: Let $x=-5$: $-2\cdot (-5)=10>8$ $✓$ Also check that $x=-4$ does NOT work: $-2\cdot (-4)=8$, and $8>8$ is false $✓$ Solution: $x\in (-\infty ,-4)$

Example 4b

Solve: $-x\le 6$

Multiply both sides by $-1$. Since $-1$ is negative, flip the sign:

Check: Let $x=0$: $-0=0\le 6$ $✓$ Solution: $x\in [-6,\infty )$

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Type 5: Two Operations Combined

These require two steps: first add/subtract, then multiply/divide.

Example 5a

Solve: $2x+5>13$

Step 1: Subtract 5 from both sides Step 2: Divide by 2 (positive, direction unchanged) Check: Let $x=5$: $2\cdot 5+5=15>13$ $✓$ Solution: $x\in (4,\infty )$

Example 5b

Solve: $-4x+3\ge 19$

Step 1: Subtract 3 from both sides Step 2: Divide by $-4$ (negative, reverse the sign!) Check: Let $x=-5$: $-4\cdot (-5)+3=20+3=23\ge 19$ $✓$ Check boundary $x=-4$: $-4\cdot (-4)+3=16+3=19\ge 19$ $✓$ Solution: $x\in (-\infty ,-4]$

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Summary

Golden rule: The inequality sign flips only when you multiply or divide by a negative number. Addition and subtraction never change the direction.

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Practice Exercises

• Inequalities - addition and subtraction - Simple one-step inequalities
• Inequalities - Number Line - Represent your solutions graphically
• Inequalities - Intervals - Write solutions as intervals

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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 - Next level of difficulty
• Linear Inequalities - Number Line - Graphical solutions