Q: How do I solve the three-part form a < x < b?

Apply every operation to all three parts. -1 < 2x + 3 < 9: subtract 3 from each part, then divide each by 2.

Q: When does an 'and' compound have no solution?

When the two pieces don't overlap, like x > 5 and x < 2. The answer is the empty set ∅.

Q: When does an 'or' compound cover all real numbers?

When the two pieces together cover the line, like x < 4 or x ≥ 1. The answer is (-∞, ∞).

Q: Do the symbols still flip when dividing by a negative?

Yes. In the three-part form, dividing all three parts by a negative reverses both symbols; then rewrite so the smaller number is on the left.

Q: How is the answer written?

An 'and' result is one interval like (-2, 3]. An 'or' result is a union: (-∞, -1) ∪ (4, ∞).

Q: How do I type an "and" answer versus an "or" answer?

An "and" result is one interval, so type it as [-2, 3) or as the inequality -2 <= x < 3. An "or" result is two pieces joined by a union: (-inf,-3) U (4,inf). When the two conditions never overlap, type no solution.

Question 1

What is the difference between 'and' and 'or' compounds?

Accepted Answer

'And' (intersection) is satisfied only by values that make both inequalities true — the overlap. 'Or' (union) is satisfied by values that make at least one true — everything either piece covers.

Question 2

How do I solve the three-part form a < x < b?

Accepted Answer

Apply every operation to all three parts. -1 < 2x + 3 < 9: subtract 3 from each part, then divide each by 2.

Question 3

When does an 'and' compound have no solution?

Accepted Answer

When the two pieces don't overlap, like x > 5 and x < 2. The answer is the empty set ∅.

Question 4

When does an 'or' compound cover all real numbers?

Accepted Answer

When the two pieces together cover the line, like x < 4 or x ≥ 1. The answer is (-∞, ∞).

Question 5

Do the symbols still flip when dividing by a negative?

Accepted Answer

Yes. In the three-part form, dividing all three parts by a negative reverses both symbols; then rewrite so the smaller number is on the left.

Question 6

How is the answer written?

Accepted Answer

An 'and' result is one interval like (-2, 3]. An 'or' result is a union: (-∞, -1) ∪ (4, ∞).

Question 7

How do I type an "and" answer versus an "or" answer?

Accepted Answer

An "and" result is one interval, so type it as [-2, 3) or as the inequality -2 <= x < 3. An "or" result is two pieces joined by a union: (-inf,-3) U (4,inf). When the two conditions never overlap, type no solution.

Compound Inequalities Practice Problems