Double inequalities

A double inequality such as −3 ≤ 2x − 5 < 7 holds two conditions at once. Solve it by doing the same step to all three parts.

Worked example: −3 ≤ 2x − 5 < 7. Add 5 to all parts: 2 ≤ 2x < 12. Divide all parts by 2: 1 ≤ x < 6, that is, [1, 6). Check: x = 1 gives −3 (allowed) and x = 6 gives 7 (not allowed).

Worked example with a reversal: 1 < (7 − 2x)/3 ≤ 3. Multiply by 3: 3 < 7 − 2x ≤ 9. Subtract 7: −4 < −2x ≤ 2. Divide by −2, reversing both signs: 2 > x ≥ −1, which reads −1 ≤ x < 2.

After a reversal, rewrite the answer with the smaller number on the left, and carry each end's type (strict or slack) with that end.

A double inequality can pass through a formula. If a cooler must stay between 4 °C and 10 °C and F = (9/5)C + 32, then (9/5)(4) + 32 < F < (9/5)(10) + 32, that is 39.2 < F < 50.

This works because F = (9/5)C + 32 increases with C, so the order of the three parts is kept.