Solving Compound Inequalities: Step-by-Step Guide
A compound inequality puts two inequalities together with AND or OR. With AND, both parts must be true at the same time. It is often written as a chain, like 2 < x < 5. With OR, at least one of the parts must be true. The answer is where the two solutions overlap (AND) or where they are combined (OR).
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Decide if it is AND or OR. A chain like a < ax + b < c means AND. Two separate inequalities joined by the word 'or' mean OR.\[a < f(x) < b\;\text{(AND)},\quad f(x) < a \text{ or } f(x) > b\;\text{(OR)}\]
- Solve each part. For an AND chain, do the same thing to all THREE parts. For an OR, solve each inequality on its own.\[a-k < f(x) - k < b - k\]
- Watch for negatives. If you multiply or divide by a negative number, flip every inequality sign.\[\text{multiply by }-1\;\Rightarrow\;\text{flip both}\]
- Put the solutions together. AND: keep only the numbers that work in both. OR: keep all the numbers that work in either one.\[\text{AND: }A\cap B,\quad \text{OR: }A\cup B\]
- Write the answer and check. Write it in interval notation. Test a number that is inside your answer and one that is outside.\[[a, b],\;(-\infty, a)\cup(b, \infty)\]
Worked example
Solve and write in interval notation: -3 <= 2x - 1 < 5
Step 1: This is a chain, so it is an AND inequality.
Step 2: Add 1 to all three parts: -3 + 1 <= 2x - 1 + 1 < 5 + 1, which gives -2 <= 2x < 6.
Step 3: Divide all three parts by 2. The 2 is positive, so nothing flips: -1 <= x < 3.
Step 4: So x is between -1 (included) and 3 (not included).
Step 5: In interval notation: [-1, 3).
Check: Try x = 0: -3 <= 2(0) - 1 = -1 < 5, which is true. Try x = 3: 2(3) - 1 = 5, which is not less than 5, so it is correctly left out.
Tips and common mistakes
- Whatever you do to one part of an AND chain, do to all three parts.
- If you multiply or divide by a negative number, every inequality sign in the chain flips.
- AND answers are usually one interval between two numbers, like [a, b].
- OR answers are often two pieces going in opposite directions, like (-infinity, a) and (b, infinity).
- If the two parts of an AND do not overlap, there is no solution.
- Brackets [ ] mean the endpoint is included. Parentheses ( ) mean it is not.
- Write chains with the smaller number on the left, so a < x < b.