9. Compound Inequalities

A compound inequality joins two inequalities.

AND means the number must satisfy both conditions:
\[ 2This means \(x\) is greater than 2 and at most 8.

OR means either condition can work:
\[ x<2 \quad \text{or} \quad x>5. \]
This means numbers below 2 and numbers above 5 are included, with a gap between.

Example 1:
\[ 2Subtract 1 from all three parts:
\[ 1

Example 2:
\[ -3\le2x+1<9. \]
Subtract 1 from all three parts:
\[ -4\le2x<8. \]
Divide all three parts by 2:
\[ -2\le x<4. \]

Example 3:
\[ -1<\frac{x-2}{3}\le4. \]
Multiply all parts by 3:
\[ -3Add 2:
\[ -1

Common mistake: solving only one side of an AND inequality. You must apply the operation to all three parts.