How to Solve: Compound Inequalities

How to Solve: Compound Inequalities

Introduction

"Mastering compound inequalities means you can answer questions like: ‘What’s the safe speed range for a car on a wet road?’ or ‘How many hours can you work without exceeding your budget?’—and nail those 5-mark exam questions every time."

What You Need To Know First

Before diving into compound inequalities, ensure you understand: 1. Basic inequalities (e.g., x > 3, y ≤ -2) and how to solve them. 2. Number line representation (open/closed circles, shading left/right). 3. Solving linear equations (e.g., 2x + 5 = 11).

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

Key Vocabulary

Formulas To Know

No formulas to memorize, but two rules to apply:

1. "And" Inequality Rule
2. Solve each inequality separately.
3. The solution is the overlap (intersection) of both solutions.

4. Example: x > 2 and x < 5 → 2 < x < 5.

5. "Or" Inequality Rule

6. Solve each inequality separately.
7. The solution is the combination (union) of both solutions.
8. Example: x < -1 or x ≥ 3 → x < -1 or x ≥ 3.

Step-by-Step Method

Follow these steps exactly for any compound inequality.

Step 1: Identify the Type

• Look for the word "and" or "or" (or a comma in some cases).
• "x > 2 and x < 5" → "and" (intersection).
• "x < -1 or x ≥ 3" → "or" (union).

Step 2: Split the Inequality

• If "and", write two separate inequalities.
• 3 < x + 2 ≤ 8 → 3 < x + 2 and x + 2 ≤ 8.
• If "or", write two separate inequalities.
• x - 1 < -4 or x + 3 ≥ 5 → x - 1 < -4 or x + 3 ≥ 5.

Step 3: Solve Each Inequality

• Solve like a normal linear inequality.
• 3 < x + 2 → x > 1.
• x + 2 ≤ 8 → x ≤ 6.
• x - 1 < -4 → x < -3.
• x + 3 ≥ 5 → x ≥ 2.

Step 4: Combine the Solutions

• "And": Find the overlap (where both are true).
• x > 1 and x ≤ 6 → 1 < x ≤ 6.
• "Or": Combine all possible solutions.
• x < -3 or x ≥ 2 → x < -3 or x ≥ 2.

Step 5: Graph the Solution (If Required)

• "And": Shade the overlapping region on a number line.
• 1 < x ≤ 6 → Open circle at 1, closed circle at 6, shade in between.
• "Or": Shade both regions separately.
• x < -3 or x ≥ 2 → Shade left of -3 (open circle) and right of 2 (closed circle).

Step 6: Write the Final Answer

• Use inequality notation or interval notation.
• 1 < x ≤ 6 → (1, 6].
• x < -3 or x ≥ 2 → (-∞, -3) ∪ [2, ∞).

Worked Example Using the Steps

Problem: Solve 2 ≤ 3x - 1 < 8.

Step 1: Identify the type. - This is an "and" inequality (no "or" word, but it’s a range).

Step 2: Split the inequality. - 2 ≤ 3x - 1 and 3x - 1 < 8.

Step 3: Solve each part. - 2 ≤ 3x - 1 → 3 ≤ 3x → x ≥ 1. - 3x - 1 < 8 → 3x < 9 → x < 3.

Step 4: Combine the solutions. - x ≥ 1 and x < 3 → 1 ≤ x < 3.

Step 5: Graph (if needed). - Closed circle at 1, open circle at 3, shade in between.

Step 6: Final answer. - Inequality: 1 ≤ x < 3. - Interval: [1, 3).

Worked Examples

Example 1 - Basic

Problem: Solve x + 4 > 1 and x - 2 < 3.

Step 1: "And" inequality. Step 2: Split into x + 4 > 1 and x - 2 < 3. Step 3: - x + 4 > 1 → x > -3. - x - 2 < 3 → x < 5. Step 4: Overlap → -3 < x < 5. Step 5: Graph: Open circles at -3 and 5, shade in between. Step 6: Final answer: (-3, 5).

What we did and why: We solved each inequality separately, then found where both were true. The overlap is the solution.

Example 2 - Medium

Problem: Solve -2x + 5 ≤ 11 or 4x - 1 > 15.

Step 1: "Or" inequality. Step 2: Split into -2x + 5 ≤ 11 or 4x - 1 > 15. Step 3: - -2x + 5 ≤ 11 → -2x ≤ 6 → x ≥ -3 (remember to flip the inequality when dividing by a negative!). - 4x - 1 > 15 → 4x > 16 → x > 4. Step 4: Combine → x ≥ -3 or x > 4. - Since x > 4 is already included in x ≥ -3, the solution is x ≥ -3. Step 5: Graph: Closed circle at -3, shade right. Step 6: Final answer: [-3, ∞).

What we did and why: We solved each part, then combined them. Since x > 4 is a subset of x ≥ -3, the simpler x ≥ -3 covers all cases.

Example 3 - Exam Style

Problem: A company’s profit P (in thousands) must satisfy 50 ≤ 2P + 10 < 90 to avoid losses. Find the range of P.

Step 1: "And" inequality (range). Step 2: Split into 50 ≤ 2P + 10 and 2P + 10 < 90. Step 3: - 50 ≤ 2P + 10 → 40 ≤ 2P → 20 ≤ P. - 2P + 10 < 90 → 2P < 80 → P < 40. Step 4: Overlap → 20 ≤ P < 40. Step 5: Graph: Closed circle at 20, open circle at 40, shade in between. Step 6: Final answer: [20, 40).

What we did and why: We treated the compound inequality as two parts, solved for P, and found the overlapping range where the company avoids losses.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Compound inequalities are just two inequalities in one. If you see ‘and,’ solve both and find the overlap—like a Venn diagram. If you see ‘or,’ solve both and combine all possible answers. Always split them first, solve separately, then bring them back together. Watch out for negative signs—they flip the inequality. And if it’s a range like 2 < x ≤ 5, treat it as two parts. Graph it to double-check, and write your final answer in both inequality and interval notation. You’ve got this—go crush that exam!