Compound Inequalities

And and or do opposite jobs. And means a number has to satisfy both conditions at once, so you're intersecting two sets and the result shrinks to whatever overlaps. Or means a number just needs to satisfy one condition, so you're combining two sets and the result spreads wider, often into two pieces that never touch. That's why -10.5 < x and x ≤ 20.3 becomes a single shaded band on a number line: every point in it is greater than -10.5 AND at most 20.3, all at the same time.

To solve a double inequality like -7 < 2x + 1 ≤ 9, treat it as three parallel strips and work all of them together. Subtract 1 from all three parts: -8 < 2x ≤ 8. Divide all three by 2: -4 < x ≤ 4. Every operation you do to the middle must happen to the left side and the right side too, because you're keeping three expressions in the same order relative to each other. The answer is one connected interval, from -4 (open) to 4 (closed).

An or statement, like x < -3 or x > 5, can never collapse into one double inequality because a double inequality only describes a single connected band — it needs the middle expression to sit between two numbers that actually overlap. Here the two rays point away from each other with a gap between them, so no number can be squeezed in the middle. This connects to graphing and interval notation: and-statements graph as one segment and use a single interval like (-4, 4], while or-statements graph as two separate rays and need union notation, like (-infinity, -3) union (5, infinity).