• Solutions: 5, 6, 7, 100, 4.1, etc. (infinitely many!)

Key Difference: Inequalities have a range of solutions!

Solving One-Step Inequalities

Solve inequalities just like equations - use inverse operations!

Addition and Subtraction

Example 1: Solve x + 5 > 12

Step 1: Subtract 5 from both sides x + 5 - 5 > 12 - 5 x > 7

Answer: x > 7

Meaning: Any number greater than 7 is a solution!

• Test: x = 8 → 8 + 5 = 13 > 12 ✓
• Test: x = 10 → 10 + 5 = 15 > 12 ✓

Example 2: Solve y - 3 ≤ 8

Step 1: Add 3 to both sides y - 3 + 3 ≤ 8 + 3 y ≤ 11

Answer: y ≤ 11

Meaning: Any number less than or equal to 11 works!

Multiplication and Division (Positive Numbers)

Example 3: Solve 4x < 20

Step 1: Divide both sides by 4 4x ÷ 4 < 20 ÷ 4 x < 5

Answer: x < 5

Example 4: Solve x/3 ≥ 6

Step 1: Multiply both sides by 3 3 × (x/3) ≥ 3 × 6 x ≥ 18

Answer: x ≥ 18

THE SPECIAL RULE: Multiplying or Dividing by Negatives

CRITICAL: When you multiply or divide both sides by a NEGATIVE number, you must FLIP the inequality symbol!

Why? Think about it: 5 > 2 is true

• Multiply both by -1: -5 ? -2
• On a number line, -5 is LEFT of -2, so -5 < -2
• The inequality flipped!

Example 1: Dividing by a Negative

Solve: -3x > 12

Step 1: Divide both sides by -3 AND flip the symbol -3x ÷ (-3) < 12 ÷ (-3) (> becomes <) x < -4

Answer: x < -4

Check: Try x = -5

• -3(-5) = 15 > 12 ✓

Example 2: Multiplying by a Negative

Solve: -x/2 ≤ 4

Step 1: Multiply both sides by -2 AND flip the symbol (-2) × (-x/2) ≥ (-2) × 4 (≤ becomes ≥) x ≥ -8

Answer: x ≥ -8

Example 3: Watch the Sign!

Solve: -5y < 25

Step 1: Divide by -5, flip the symbol -5y ÷ (-5) > 25 ÷ (-5) y > -5

Answer: y > -5

Remember: ONLY flip when multiplying or dividing by a NEGATIVE!

Solving Multi-Step Inequalities

Use the same strategy as multi-step equations!

Example 1: Two Steps

Solve: 3x + 7 < 22

Step 1: Subtract 7 from both sides 3x < 15

Step 2: Divide by 3 x < 5

Answer: x < 5

Example 2: With Negative Coefficient

Solve: -2x + 5 ≥ 13

Step 1: Subtract 5 from both sides -2x ≥ 8

Step 2: Divide by -2 AND FLIP x ≤ -4

Answer: x ≤ -4

Example 3: Distributive Property

Solve: 3(x - 4) > 15

Step 1: Distribute 3x - 12 > 15

Step 2: Add 12 to both sides 3x > 27

Step 3: Divide by 3 x > 9

Answer: x > 9

Example 4: Variable on Both Sides

Solve: 5x - 3 < 2x + 9

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

Step 2: Add 3 to both sides 3x < 12

Step 3: Divide by 3 x < 4

Answer: x < 4

Graphing Inequalities on a Number Line

Symbols and Graphing:

• < or > use an open circle (○) - the number is NOT included
• ≤ or ≥ use a closed/filled circle (●) - the number IS included
• Shade to the left for "less than"
• Shade to the right for "greater than"

Example 1: x > 3

Graph: Open circle at 3, shade RIGHT

• 3 is NOT included (open circle)
• All numbers greater than 3 are shaded

Example 2: x ≤ -2

Graph: Closed circle at -2, shade LEFT

• -2 IS included (closed circle)
• All numbers less than or equal to -2 are shaded

Example 3: x ≥ 0

Graph: Closed circle at 0, shade RIGHT

• 0 IS included
• All positive numbers and zero are shaded

Writing Inequalities from Graphs

Look at the circle (open or closed) and the shading direction!

Example 1: Closed circle at 5, shaded right

• Answer: x ≥ 5

Example 2: Open circle at -3, shaded left

• Answer: x < -3

Example 3: Closed circle at 0, shaded left

• Answer: x ≤ 0

Writing Inequalities from Word Problems

Translate words into inequality symbols!

Word Clues:

Greater than (>):

• "more than"
• "exceeds"
• "above"

Greater than or equal to (≥):

• "at least"
• "no less than"
• "minimum"

Less than (<):

• "fewer than"
• "below"
• "under"

Less than or equal to (≤):

• "at most"
• "no more than"
• "maximum"

Example 1: At Least

"You must be at least 13 years old to have a social media account."

Inequality: a ≥ 13 (age must be greater than or equal to 13)

Example 2: No More Than

"The elevator can hold no more than 2,000 pounds."

Inequality: w ≤ 2,000 (weight must be less than or equal to 2,000)

Example 3: More Than

"Sarah has more than $50 in her account."

Inequality: m > 50 (money is greater than 50)

Real-World Applications

Shopping with a Budget

Problem: You have $100 to spend. You buy a shirt for$25. If jeans cost $35 each, how many pairs can you buy?

Let x = number of jeans

Inequality: 25 + 35x ≤ 100

Solution: 35x ≤ 75 x ≤ 75/35 x ≤ 2.14...

Answer: You can buy at most 2 pairs of jeans (since you can't buy a fraction of jeans!)

Grade Requirements

Problem: Your test average must be at least 80% to get a B. You've taken 3 tests and scored 75, 82, and 78. What do you need on the 4th test?

Let x = 4th test score

Inequality: (75 + 82 + 78 + x)/4 ≥ 80

Solution: (235 + x)/4 ≥ 80 235 + x ≥ 320 x ≥ 85

Answer: You need at least 85% on the 4th test

Speed Limits

Problem: The speed limit is 65 mph. Write an inequality for legal speeds.

Let s = speed

Inequality: s ≤ 65

Weight Restrictions

Problem: A bridge has a maximum weight of 10 tons. A truck weighs 3 tons. If each crate weighs 0.5 tons, how many crates can the truck carry?

Let c = number of crates

Inequality: 3 + 0.5c ≤ 10

Solution: 0.5c ≤ 7 c ≤ 14

Answer: Maximum 14 crates

Compound Inequalities (Introduction)

Sometimes a value is between two numbers!

Example: "Temperature should be between 60°F and 75°F"

Written as: 60 ≤ t ≤ 75 or 60 < t < 75

This means t ≥ 60 AND t ≤ 75 at the same time.

Graph: Closed circles at 60 and 75, shade between them

Common Mistakes to Avoid

❌ Mistake 1: Forgetting to flip when dividing by negative

• Wrong: -2x > 8 → x > -4
• Right: -2x > 8 → x < -4 (flip!)

❌ Mistake 2: Flipping when you shouldn't

• Wrong: 3x < 12 → x > 4 (no negative, don't flip!)
• Right: 3x < 12 → x < 4

❌ Mistake 3: Using wrong circle on graph

• < or > : open circle ○
• ≤ or ≥ : closed circle ●

❌ Mistake 4: Shading wrong direction

• < or ≤ : shade LEFT

• or ≥ : shade RIGHT

❌ Mistake 5: Treating inequality like equation

• Inequalities have MANY solutions, not just one!

Practice Strategy

Step 1: Solve like an equation

• Use inverse operations
• Isolate the variable

Step 2: Watch for negatives!

• If you multiply or divide by a negative, FLIP the symbol

Step 3: Graph your answer

• Choose correct circle (open or closed)
• Shade correct direction

Step 4: Check with a test value

• Pick a number that should work
• Substitute it in
• Verify it makes the inequality true

Checking Your Solution

Example: Solve x + 5 < 12, got answer x < 7

Check 1: Try x = 6 (should work)

• 6 + 5 = 11 < 12 ✓ True!

Check 2: Try x = 7 (boundary, should NOT work since it's <, not ≤)

• 7 + 5 = 12 NOT < 12 ✓ Correct!

Check 3: Try x = 8 (should NOT work)

• 8 + 5 = 13 NOT < 12 ✓ Correct!

Quick Reference Chart

Graphing:

• < or > : Open circle, shade direction
• ≤ or ≥ : Closed circle, shade direction

Word Clues:

• At least, minimum, no less than → ≥
• At most, maximum, no more than → ≤
• More than, above, exceeds → >
• Less than, below, under → <

Summary

Inequalities show relationships using <, >, ≤, ≥ symbols.

Solving:

• Use inverse operations like equations
• FLIP the symbol when multiplying or dividing by a negative
• Solution is a range of values, not just one number

Graphing:

• Open circle (○) for < or >
• Closed circle (●) for ≤ or ≥
• Shade left for "less than"
• Shade right for "greater than"

Applications:

• Budgets (at most, no more than)
• Requirements (at least, minimum)
• Limits (maximum, speed limits)
• Constraints (weight, capacity)

Master inequalities and you're ready for systems of inequalities, linear programming, and real-world optimization!