Inequalities: Graphing Solutions

Inequalities represent ranges of values, not just single solutions!

INEQUALITY SYMBOLS: - < means "less than" - > means "greater than" - ≤ means "less than or equal to" - ≥ means "greater than or equal to"

KEY DIFFERENCE FROM EQUATIONS: - Equation: x = 5 → ONE solution - Inequality: x > 5 → INFINITE solutions (all numbers greater than 5!)

SOLVING ONE-STEP INEQUALITIES:

Addition/Subtraction: Works just like equations!

Example: x + 3 < 10 Subtract 3: x < 7

Example: x - 5 ≥ 2 Add 5: x ≥ 7

Multiplication/Division by POSITIVE: Works just like equations!

Example: 3x < 12 Divide by 3: x < 4

Example: x/2 > 6 Multiply by 2: x > 12

Multiplication/Division by NEGATIVE: CRITICAL RULE: FLIP THE INEQUALITY SIGN!

Example: -2x < 8 Divide by -2: x > -4 (flipped from < to >)

Example: -x ≥ 5 Multiply by -1: x ≤ -5 (flipped from ≥ to ≤)

WHY FLIP? Because multiplying by negative reverses order! Think: 3 < 5, but -3 > -5

GRAPHING ON A NUMBER LINE:

Step 1: Draw a number line Mark important values including the boundary number

Step 2: Mark the boundary point - Use OPEN CIRCLE (○) for < or > (not including the number) - Use CLOSED CIRCLE (●) for ≤ or ≥ (including the number)

Step 3: Shade the solution region - Shade RIGHT for > or ≥ (greater = right) - Shade LEFT for < or ≤ (less = left)

GRAPHING EXAMPLES:

Example 1: x < 3
- Open circle at 3 (not including 3)
- Shade left (all numbers less than 3)
Graph: ←●────3────→

Example 2: x ≥ -2
- Closed circle at -2 (including -2)
- Shade right (all numbers -2 and greater)
Graph: ←────(-2)●→

Example 3: x > 0
- Open circle at 0
- Shade right (positive numbers)
Graph: ←────0○──→

Example 4: x ≤ 5
- Closed circle at 5
- Shade left (5 and all smaller numbers)
Graph: ←──●5────→

WRITING INEQUALITIES FROM GRAPHS:

Look for: 1. Circle type: Open (< or >) vs Closed (≤ or ≥) 2. Shading direction: Left (less) vs Right (greater)

Example: Closed circle at 4, shaded right Answer: x ≥ 4

Example: Open circle at -1, shaded left Answer: x < -1

CHECKING SOLUTIONS: Pick a number from the shaded region and test!

Example: If x < 5, test x = 3 3 < 5? YES ✓ (3 is in the shaded region)

Test x = 7 7 < 5? NO (7 is not in the shaded region)

REAL-WORLD INEQUALITIES: - Age restrictions: "Must be at least 13" → a ≥ 13 - Speed limits: "Under 65 mph" → s < 65 - Budget: "Spend no more than $50" → c ≤ 50 - Minimum height: "Must be taller than 48 inches" → h > 48 - Temperature: "Keep below 75°F" → t < 75

Master graphing inequalities on number lines!

GRAPH THESE INEQUALITIES: For 1-8, draw a number line and graph the solution.

1. x > 4 2. x ≤ -2 3. x < 0 4. x ≥ 7 5. x > -3 6. x ≤ 5 7. x < -1 8. x ≥ 0

SOLVE AND GRAPH: Solve the inequality, then graph the solution.

9. x + 3 > 7 10. x - 5 ≤ 2 11. 4x < 16 12. x/3 ≥ 2 13. -2x > 6 (Remember to flip!) 14. -x ≤ 4 (Remember to flip!) 15. x + 8 < 5 16. 3x ≥ -9

WRITE THE INEQUALITY FROM THE GRAPH: For 17-20, write the inequality shown by the graph.

17. Open circle at 3, shaded right 18. Closed circle at -1, shaded left 19. Closed circle at 5, shaded right 20. Open circle at 0, shaded left

WORD PROBLEMS: Write and solve the inequality, then graph.

21. A movie theater requires patrons to be at least 13 years old for a PG-13 movie. Write and graph an inequality for allowable ages.

22. A bridge has a weight limit of 10,000 pounds. Write and graph an inequality for safe weights.

23. To ride a roller coaster, you must be taller than 48 inches. Write and graph the inequality.

24. The temperature must stay below 32°F for snow. Write and graph the inequality.

CHECK SOLUTIONS: 25. Is x = 5 a solution to x < 8? 26. Is x = -3 a solution to x > -3? 27. Is x = 0 a solution to x ≤ 0? 28. Is x = 7 a solution to 2x + 1 < 16?

CHALLENGE: 29. Graph x ≥ -4 AND x < 2 on the same number line.

30. Write an inequality that has these three values as solutions: -5, 0, 3. Graph it.