How to Solve Inequalities: Complete Guide

How to Solve Inequalities: Complete Guide

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Mastering inequalities means you can answer: ‘How many hours can I work without going over my budget?’ or ‘What’s the minimum score I need to pass?’—questions that show up on every algebra exam!

What You Need To Know First

Before tackling inequalities, you must already understand: 1. Solving linear equations (e.g., 3x + 5 = 11). 2. Number line basics (e.g., plotting x > 2). 3. Operations with negative numbers (e.g., -3 × -4 = 12).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

(All are MEMORISE THIS unless marked otherwise.)

1. Basic Inequality Properties

2. If a < b, then:

   • a + c < b + c (add same number to both sides).
   • a - c < b - c (subtract same number from both sides).
   • a × c < b × c only if c > 0 (multiply by positive).
   • a × c > b × c if c < 0 (flip sign when multiplying by negative).
   • a ÷ c < b ÷ c only if c > 0 (divide by positive).
   • a ÷ c > b ÷ c if c < 0 (flip sign when dividing by negative).

3. Compound Inequality (AND/OR)

4. "x > 2 AND x ≤ 5" → 2 < x ≤ 5 (written as one statement).
5. "x < -1 OR x ≥ 3" → Two separate intervals.

Step-by-Step Method

How to Solve Any Inequality (Linear or Compound)

1. Isolate the variable term (get x by itself on one side).
2. Use addition/subtraction first.

3. Example: 3x + 5 < 11 → Subtract 5: 3x < 6.

4. Solve for the variable (divide or multiply).

5. If multiplying/dividing by a negative, flip the inequality sign.

6. Example: -2x > 8 → Divide by -2: x < -4.

7. Write the solution in inequality form (e.g., x > 3).

8. For compound inequalities, keep the variable in the middle (e.g., -1 ≤ x < 4).

9. Graph the solution on a number line (if required).

10. Use an open circle for > or <.
11. Use a closed circle for ≥ or ≤.

12. Shade the region where solutions lie.

13. Check your answer (plug in a number from your solution set).

14. Example: For x > 2, test x = 3: 3 > 2 ✔️.

Worked Example Using the Steps

Problem: Solve -4x + 7 ≤ 15.

1. Isolate the variable term:
   -4x + 7 ≤ 15
   Subtract 7: -4x ≤ 8.

2. Solve for x:
   Divide by -4 (flip the sign!): x ≥ -2.

3. Write the solution:
   x ≥ -2.

4. Graph on a number line:

5. Closed circle at -2.

6. Shade to the right (all numbers ≥ -2).

7. Check:
   Test x = 0: -4(0) + 7 = 7 ≤ 15 ✔️.

Worked Examples

Example 1 - Basic

Problem: Solve 5x - 3 > 12.

Solution: 1. Add 3 to both sides: 5x > 15. 2. Divide by 5: x > 3. 3. Graph: Open circle at 3, shade right. 4. Check: Test x = 4: 5(4) - 3 = 17 > 12 ✔️.

What we did and why: - We isolated x by undoing the operations (subtraction first, then division). - No sign flip was needed because we divided by a positive number.

Example 2 - Medium (Negative Coefficient)

Problem: Solve -2(x + 4) ≥ 10.

Solution: 1. Distribute the -2: -2x - 8 ≥ 10. 2. Add 8: -2x ≥ 18. 3. Divide by -2 (flip the sign!): x ≤ -9. 4. Graph: Closed circle at -9, shade left. 5. Check: Test x = -10: -2(-10 + 4) = -2(-6) = 12 ≥ 10 ✔️.

What we did and why: - We had to flip the sign because we divided by a negative number. - Always distribute first before isolating x.

Example 3 - Exam Style (Compound Inequality)

Problem: Solve 3 ≤ 2x + 1 < 9.

Solution: 1. Subtract 1 from all parts: 2 ≤ 2x < 8. 2. Divide by 2: 1 ≤ x < 4. 3. Graph: Closed circle at 1, open circle at 4, shade between. 4. Check:
- Test x = 2: 3 ≤ 5 < 9 ✔️.
- Test x = 4: 3 ≤ 9 < 9 ❌ (4 is not < 4, so correct).

What we did and why: - This is an "and" inequality, so we solved all parts at once. - We kept the variable in the middle to avoid splitting into two inequalities.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Here’s the 60-second version:

1. Solve inequalities like equations—add, subtract, multiply, divide—but flip the sign when multiplying or dividing by a negative.
2. Graph carefully: Open circle for > or <, closed for ≥ or ≤. Shade the correct side.
3. Compound inequalities? If it’s and, keep it as one statement (2 < x ≤ 5). If it’s or, split it into two (x < -1 or x > 3).
4. Always check your answer—pick a number from your solution and plug it back in.
5. Watch for traps: Negative signs, hidden or/and, and disguised compound inequalities.

You’ve got this. Now go practice—start with the basic ones, then tackle the exam-style questions. And remember: flip the sign, graph it right, and check your work. Good luck!