What Are Compound Inequalities? 🤔
A compound inequality is a sentence with two inequality statements joined by either the word "AND" or the word "OR". They are useful for describing a range of values that satisfy multiple conditions at once, like a number being between two other numbers.
"AND" means a solution must satisfy both inequalities. "OR" means a solution must satisfy at least one of the inequalities.
How to Solve Compound Inequalities
- Identify the Joining Word: Is it "AND" or "OR"? This determines your solution set.
- Solve Each Inequality Separately: Isolate the variable in each part, just like a regular inequality.
- Combine the Solutions:
- For "AND": Find the overlap where both solutions are true. Graphically, this is where the two number lines overlap.
- For "OR": Combine all solutions from both inequalities. Graphically, you put both sets on the number line.
- Graph the Solution: Represent your final answer on a number line to visualize it.
Worked Examples
Example 1: "AND" Inequality
Solve and graph: -2 ≤ 3x + 1 < 7
- Subtract 1 from all three parts: -2 - 1 ≤ 3x + 1 - 1 < 7 - 1 → -3 ≤ 3x < 6
- Divide all three parts by 3: -3/3 ≤ 3x/3 < 6/3 → -1 ≤ x < 2
- Graph: A closed circle at -1, an open circle at 2, and shading in between.
Solution: [-1, 2)
Example 2: "OR" Inequality
Solve and graph: 2x + 3 < 1 OR x - 5 > 0
- Solve the first inequality: 2x < 1 - 3 → 2x < -2 → x < -1
- Solve the second inequality: x > 0 + 5 → x > 5
- Combine with "OR": The solution is all numbers less than -1 OR greater than 5.
- Graph: An arrow left from an open circle at -1, and an arrow right from an open circle at 5.
Solution: (-∞, -1) U (5, ∞)
Common Mistakes to Avoid ⚠️
Reversing the Inequality: Remember to flip the inequality sign only when you multiply or divide by a negative number.
Confusing "AND" and "OR": "AND" needs an overlap (like between two numbers). "OR" combines everything (like outside of two numbers).
Incorrect Graphing: Use an open circle for < or > and a closed circle for ≤ or ≥.
Tips & Tricks
The "Zappy Line" Method: For "AND" statements like -1 ≤ x < 2, draw a zappy line (or a thick line) on your number line from -1 to 2 to clearly show the included range.
Word Association: Think of "AND" as the strict guard—both conditions must be met. Think of "OR" as the flexible guard—either condition works.
How to Practice
- Start with simple "AND" problems to master solving and graphing a single range.
- Move on to "OR" problems, paying close attention to how the graph has two separate sections.
- Create your own problems and solve them, or try to write the inequality from a given number line graph.
- Use online practice tools that provide instant feedback on your steps and final answer.