Solving Compound Inequalities

What is a compound inequality?

A compound inequality has two inequality signs — it constrains $x$ to lie between two bounds simultaneously. Think of it as "a < stuff ≤ b" meaning "stuff is between $a$ and $b$."

Step-by-step: Solve $-3 < 2x + 1 \leq 9$

The goal is to isolate $x$ in the middle. Apply the same operation to all three parts.

Step 1 — Subtract 1 from all three parts:

$-3 - 1 < 2x + 1 - 1 \leq 9 - 1$ $-4 < 2x \leq 8$

Step 2 — Divide all three parts by 2:

$\frac{-4}{2} < \frac{2x}{2} \leq \frac{8}{2}$ $-2 < x \leq 4$

Step 3 — Interpret the result. $x$ is greater than $-2$ (but NOT equal, open circle) and less than or equal to $4$ (closed circle).

Step 4 — Graph on a number line: Open circle at $-2$, closed circle at $4$, shade the interval between them.

Step 5 — Interval notation: $(-2, 4]$.

Check: Try $x = 0$ (inside): $-3 < 2(0)+1 = 1 \leq 9$ ✓. Try $x = -3$ (outside): $-3 < 2(-3)+1 = -5$? No, $-3 \not< -5$ ✗. Try $x = 4$ (boundary): $-3 < 2(4)+1 = 9 \leq 9$ ✓ (included because $\leq$).

The flip rule still applies

If you multiply/divide by a negative number, flip BOTH inequality signs. For example:

$6 > -3x \geq -12 \implies -2 < x \leq 4 \quad \text{(divided by -3, flipped both)}$

Try it in the visualization

Adjust the bounds and coefficient. The number line shows the solution interval with correct open/closed circles. Test points verify which values are inside or outside the interval.