Linear Inequalities

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using inequality symbols instead of an equals sign.

Inequality Symbols:

• < less than
• > greater than
• ≤ less than or equal to
• ≥ greater than or equal to

Examples:

• x < 5 (x is less than 5)
• x ≥ 3 (x is greater than or equal to 3)
• 2x + 1 > 7 (2x + 1 is greater than 7)

Difference from Equations

Equations have ONE solution (or specific number of solutions) Example: x + 2 = 5 has solution x = 3

Inequalities have INFINITE solutions (a range of values) Example: x + 2 < 5 has solutions x < 3 (all numbers less than 3)

Solutions of Inequalities

A solution to an inequality is any value that makes the inequality true.

Example: Is x = 2 a solution to x < 5? Check: 2 < 5? Yes ✓ So x = 2 is a solution.

Is x = 7 a solution to x < 5? Check: 7 < 5? No ✗ So x = 7 is NOT a solution.

Graphing Solutions on a Number Line

We represent solutions visually on a number line:

Open Circle (○): Use for < or > The number is NOT included in the solution

Closed Circle (●): Use for ≤ or ≥ The number IS included in the solution

Arrow: Shows direction of other solutions

Examples:

x < 3 ○ at 3, arrow pointing left Numbers less than 3

x ≥ -2 ● at -2, arrow pointing right -2 and numbers greater than -2

Solving One-Step Inequalities

Solve inequalities just like equations, with ONE IMPORTANT EXCEPTION: flip the inequality sign when multiplying or dividing by a NEGATIVE number.

Addition/Subtraction (no sign flip):

Example 1: x + 4 < 9 Subtract 4 from both sides: x < 5

Example 2: x - 3 ≥ 7 Add 3 to both sides: x ≥ 10

Multiplication/Division by POSITIVE (no sign flip):

Example 3: 3x > 12 Divide both sides by 3: x > 4

Example 4: x/2 ≤ 5 Multiply both sides by 2: x ≤ 10

Multiplication/Division by NEGATIVE (FLIP the sign!):

Example 5: -2x < 8 Divide both sides by -2 AND flip sign: x > -4

Example 6: -x ≥ 5 Multiply both sides by -1 AND flip sign: x ≤ -5

Why Flip the Sign for Negatives?

When you multiply or divide by a negative, the order of numbers reverses.

Example: Start with true statement 3 < 5 Multiply both sides by -1: -3 ? -5

On a number line, -3 is to the RIGHT of -5, so: -3 > -5

The inequality flipped!

Solving Two-Step Inequalities

Use the same process as equations: undo addition/subtraction first, then multiplication/division.

Example 1: 2x + 5 < 13 Step 1: Subtract 5 from both sides 2x < 8

Step 2: Divide both sides by 2 x < 4

Example 2: -3x + 7 ≥ 16 Step 1: Subtract 7 from both sides -3x ≥ 9

Step 2: Divide both sides by -3 (FLIP sign!) x ≤ -3

Example 3: x/4 - 3 > 2 Step 1: Add 3 to both sides x/4 > 5

Step 2: Multiply both sides by 4 x > 20

Solving Multi-Step Inequalities

Follow the same steps as multi-step equations:

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: 5x - 3 < 2x + 9 Step 1: Subtract 2x from both sides 3x - 3 < 9

Step 2: Add 3 3x < 12

Step 3: Divide by 3 x < 4

Example 3: -2(x + 4) > 6 Step 1: Distribute -2 -2x - 8 > 6

Step 2: Add 8 -2x > 14

Step 3: Divide by -2 (FLIP!) x < -7

Compound Inequalities

Compound inequalities combine two inequalities.

"And" Compound Inequalities: Written as a < x < b or using AND The solution satisfies BOTH inequalities (overlap)

Example: -2 < x < 5 Read as: "x is greater than -2 AND less than 5" Solution: numbers between -2 and 5 Graph: ○ at -2, ○ at 5, line between them

Solving: -3 < 2x + 1 < 7 Solve as three parts: -3 < 2x + 1 AND 2x + 1 < 7 -4 < 2x AND 2x < 6 -2 < x AND x < 3 Solution: -2 < x < 3

"Or" Compound Inequalities: The solution satisfies EITHER inequality (union)

Example: x < -1 OR x > 3 Solution: numbers less than -1 or greater than 3 Graph: ○ at -1 with arrow left, ○ at 3 with arrow right

Graphing Linear Inequalities in Two Variables

When graphing inequalities like y > 2x + 1:

Step 1: Graph the boundary line (as if it were an equation)

• Use solid line for ≤ or ≥
• Use dashed line for < or >

Step 2: Shade the appropriate region

• For y > or y ≥: shade ABOVE the line
• For y < or y ≤: shade BELOW the line

Test Point Method: Pick a test point (often (0,0) if not on the line) Substitute into inequality If true, shade region containing that point If false, shade the other region

Example: Graph y < 2x + 1

Step 1: Graph y = 2x + 1 with DASHED line (slope 2, y-intercept 1)

Step 2: Test point (0, 0) 0 < 2(0) + 1 0 < 1 ✓ true

Step 3: Shade region containing (0, 0) (below line)

Checking Solutions

Always check by substituting a value from your solution back into the original inequality.

Example: If solving 3x - 4 > 8 gives x > 4 Check with x = 5: 3(5) - 4 > 8 15 - 4 > 8 11 > 8 ✓

Special Cases

All Real Numbers: When you get a statement that's always true Example: x + 3 > x + 1 After simplifying: 3 > 1 (always true) Solution: all real numbers

No Solution: When you get a false statement Example: x + 3 < x + 1 After simplifying: 3 < 1 (never true) Solution: no solution (empty set)

Common Mistakes to Avoid

1. Forgetting to flip the inequality sign When dividing by -2, must flip! -2x < 8 becomes x > -4, NOT x < -4

2. Using wrong circle type < or > use open circle ○ ≤ or ≥ use closed circle ●

3. Wrong direction on number line x < 3 means numbers to the LEFT of 3 x > 3 means numbers to the RIGHT of 3

4. Not distributing negative correctly -2(x - 3) = -2x + 6, not -2x - 6

5. Treating inequality like an equation Remember: infinity solutions, not just one!

Real-World Applications

Inequalities model many real situations:

Example 1: Budget "You have $50. Each movie ticket costs$12. How many tickets can you buy?" Let x = number of tickets 12x ≤ 50 x ≤ 4.17 You can buy at most 4 tickets (can't buy partial ticket)

Example 2: Speed Limit "The speed limit is 65 mph" s ≤ 65 (your speed must be at most 65)

Example 3: Minimum Age "You must be at least 16 to drive" a ≥ 16 (your age must be 16 or greater)

Example 4: Temperature "Water is liquid between 32°F and 212°F" 32 < T < 212 (temperature is between 32 and 212)

Problem-Solving Strategy

1. Read problem and identify what you're solving for
2. Define a variable
3. Identify the inequality symbol from words:
   • "at least" means ≥
   • "at most" means ≤
   • "more than" means >
   • "less than" means <
   • "no more than" means ≤
   • "no less than" means ≥
4. Write the inequality
5. Solve the inequality
6. Interpret answer in context

Key Words and Phrases

| Phrase | Symbol | |--------|--------| | Greater than | > | | Less than | < | | At least | ≥ | | At most | ≤ | | No more than | ≤ | | No less than | ≥ | | More than | > | | Fewer than | < | | Minimum | ≥ | | Maximum | ≤ |

Quick Reference

| Inequality | Read As | Number Line | |------------|---------|-------------| | x < 3 | x less than 3 | ○ at 3, arrow left | | x > 3 | x greater than 3 | ○ at 3, arrow right | | x ≤ 3 | x at most 3 | ● at 3, arrow left | | x ≥ 3 | x at least 3 | ● at 3, arrow right |

Practice Tips

• Always check if you need to flip the sign
• Test a value to verify your solution
• Draw number lines clearly with correct circles
• Remember: ≤ and ≥ include the endpoint
• Word problems: translate carefully to inequality symbols
• Keep track of negative signs when distributing