Algebra
Single-variable inequalities
Compound inequalities
20 practice questions
2 video lessons
Theory + worked examples
Theory
A compound inequality combines two inequalities:
- And (intersection): both must hold, e.g. \(-2\le x<3\) — the values between.
- Or (union): either holds, e.g. \(x<-1\) or \(x\ge4\) — two separate rays.
“And” overlaps to one segment; “or” keeps both pieces.
An “and” is the segment between.
An “or” is two rays.
The two types:
\[a<x<b\ (\text{and}),\qquad x<a\ \text{or}\ x>b\ (\text{or})\]
Solve an “and” by operating on all three parts at once.
How to solve compound inequalities
- Identify “and” or “or”.
- For “and”, solve all parts together.
- For “or”, solve each separately.
- Graph the intersection or union.
Example 1 — An 'and' (between)
Graph \(-2\le x<3\).
Solution
“And” means the overlap — the values between.
| \(-2\le x<3\) | \(\Rightarrow\) | \(\text{one segment}\) |
Example 2 — An 'or' (union)
Graph \(x<-1\) or \(x\ge4\).
Solution
“Or” means either piece — two rays.
| \(x<-1\ \text{or}\ x\ge4\) | \(\Rightarrow\) | \(\text{two rays}\) |
Example 3 — Solve a compound 'and'
Solve \(1<2x+3\le9\).
Solution
Work all three parts together.
| \(-2\) | < | \(2x\le6\) |
| \(-1\) | < | \(x\le3\) |
Example 4 — And vs or
What is the difference between an “and” and an “or” compound inequality?
Solution
“And” is the intersection (both true); “or” is the union (either true).
Common pitfalls
“And” is the overlap; “or” is the union.
Apply each step to all three parts of a between statement.
Flip both signs if multiplying a between statement by a negative.
Frequently asked questions
What is a compound inequality?
Two inequalities joined by “and” or “or.”
What does an “and” inequality look like?
A between statement like \(-2\le x<3\).
What does an “or” inequality look like?
Two separate rays, like \(x<-1\) or \(x\ge4\).
Is “and” the intersection or union?
The intersection — both must be true.
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