Questions: Compound Inequalities

Question 1

A machine is safe to operate only when the temperature is above −5°C and below 40°C. Which compound inequality correctly represents the safe operating range?

• A. T < −5 OR T > 40
• B. T > −5 OR T < 40
• C. T > −5 AND T < 40, equivalently written −5 < T < 40
• D. T < −5 AND T > 40

Answer: (C)

Both conditions must hold simultaneously — the temperature must be above −5 AND below 40. This is an 'and' (intersection) compound inequality, graphed as a segment between the two endpoints. Option A represents the unsafe shutdown range. Option B uses 'or' which would mean 'any temperature except exactly −5 or exactly 40' — almost all temperatures. Option D is logically impossible (nothing is simultaneously less than −5 and greater than 40).

Question 2

Solving a compound 'and' inequality yields x > 8 AND x < 3. What is the solution set?

• A. 3 < x < 8
• B. x < 3 OR x > 8
• C. All real numbers
• D. No solution (empty set)

Answer: (D)

No number can simultaneously be greater than 8 and less than 3. The 'and' requires both conditions to be true at once, but these two constraints are mutually exclusive — there is no overlap. This is an empty set. Option A reverses the direction of both inequalities into a valid range, which is a common error. Recognizing impossible 'and' constraints is a key skill; it reveals the underlying logic of intersection.

Question 3

An 'or' compound inequality can have a solution set that includes every real number.

• T. True
• F. False

Answer: True

If an 'or' inequality produces two overlapping or covering conditions — for example, x < 10 OR x > 2 — then every real number satisfies at least one condition (any number is either less than 10, or greater than 2, or both). The union can expand to cover the entire number line. This edge case illustrates that 'or' can only widen a solution set, never narrow it.

Question 4

An 'and' compound inequality generally has at least one solution.

• T. True
• F. False

Answer: False

An 'and' compound inequality requires both conditions to hold simultaneously — which may be impossible. For example, x > 7 AND x < 3 has no solution because no number satisfies both constraints. The intersection of two non-overlapping ranges is the empty set. This is why solving each part of an 'and' compound inequality and checking whether the regions actually overlap is a necessary step.

Question 5

Explain the difference between 'and' and 'or' compound inequalities in terms of what kind of solution set each produces.

Model answer: 'And' means both conditions must be true simultaneously — the solution set is the intersection of the two individual solution sets. On a number line, this looks like a segment between two values (or is empty if the conditions conflict). 'Or' means at least one condition must be true — the solution set is the union of the two individual solution sets. On a number line, this typically looks like two outward-pointing rays with a gap in between (or all real numbers if the regions overlap).

'And' narrows — it can only produce a solution set smaller than or equal to either individual inequality. 'Or' widens — it can only produce a solution set larger than or equal to either individual inequality. The logical connective determines the fundamental shape of the solution, which is why misreading 'and' as 'or' (or vice versa) produces a dramatically wrong answer, not just a minor error.