A compound inequality joins two inequalities with and or or. You solve each piece with the same moves as before; the new part is how the pieces combine.
Why it works
“And” means both statements must be true at once, so the solutions are where the two graphs overlap. A number between $-2$ and $3$ is greater than $-2$ and less than $3$, which is why we write it $-2 < x < 3$, one piece. “Or” means at least one statement is true, so the solutions are everything in either graph. If the pieces point away from each other, the answer stays as two separate rays.
| Words | Means | Graph | Written |
|---|---|---|---|
| $x > -2$ and $x \le 3$ | both at once | one segment | $-2 < x \le 3$ |
| $x < -2$ or $x > 5$ | either one | two rays | $x < -2$ or $x > 5$ |
Worked examples
Worked example: An “and” in three parts
Solve $-3 \le 2x + 1 < 7$.
- This says $2x + 1$ is between $-3$ and $7$. Do the same thing to all three parts.
- Subtract $1$: $-4 \le 2x < 6$.
- Divide by $2$: $-2 \le x < 3$.
Answer: $-2 \le x < 3$: a segment from $-2$ (closed) to $3$ (open).
Worked example: An “or”
Solve $x + 3 < 1$ or $2x - 1 > 9$.
- First piece: $x < -2$.
- Second piece: $2x > 10$, so $x > 5$.
- Keep every number in either piece. They do not overlap, so the answer stays as two rays.
Answer: $x < -2$ or $x > 5$.
Worked example: Dividing by a negative
Solve $-5 < 1 - 2x \le 7$.
- Subtract $1$ from all three parts: $-6 < -2x \le 6$.
- Divide all three parts by $-2$ and flip both signs: $3 > x \ge -3$.
- Rewrite from smallest to largest: $-3 \le x < 3$.
Answer: $-3 \le x < 3$
Common mistake
Writing an “or” answer like $x < -2$ or $x > 5$ as $5 < x < -2$. That chain claims $5 < -2$, so no number satisfies it. A chained inequality always means “and,” with the smaller number on the left. Two rays pointing apart must be written as two pieces joined by “or.”
Key idea
“And” keeps only the overlap and is written as one chain, smallest number first. “Or” keeps everything in either piece and is written as two inequalities joined by “or.”