Overview
An AND condition requires both inequalities and describes their intersection. An OR condition accepts either inequality and describes their union. A chained comparison contains two conditions that can be transformed together when the same positive operation applies to all three parts.
Consider this example: Solve −2 < x + 3 ≤ 7. Subtract 3 from all three parts, giving −5 < x ≤ 4. The lower endpoint stays excluded and the upper endpoint stays included.
The method also handles a different case: Solve 1 ≤ 2x − 3 < 9. Add 3 throughout: 4 ≤ 2x < 12. Divide throughout by 2 to get 2 ≤ x < 6. Dividing only the center would leave incorrect boundaries.
For a further application, consider: How many integers x satisfy −6 < −2x ≤ 4 and x ≠ 0? Dividing the chain by −2 reverses both comparisons: 3 > x ≥ −2. Thus the integer candidates are −2, −1, 0, 1, 2. The additional restriction removes 0, leaving 4 values. Including 3 would violate the strict lower comparison on −2x.
Using a union for AND admits values that meet only one requirement. Changing just the center of a chained comparison breaks the relationship. For a new problem, decide whether all conditions or any condition must hold. Test boundary values, keeping open and closed endpoints consistent with the signs.