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Interactive practice · 16 questions on compound inequalities — AND (intersection) vs OR (union), solving three-part inequalities (including the negative-coefficient flip), interval notation with unions, and the no-solution / all-real special cases — with step-by-step solutions.
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Easy Which describes the solution of \(x \ge 2\) AND \(x < 7\)?
Answer: C — \(2 \le x < 7\); interval \([2,\,7)\)
Easy Which describes the solution of \(x < -1\) OR \(x > 4\)?
Answer: A — \((-\infty,\,-1) \cup (4,\,\infty)\)
Easy For an AND (conjunction) compound inequality, a value is a solution when it satisfies:
Answer: A — both inequalities (the intersection)
Easy For an OR (disjunction) compound inequality, a value is a solution when it satisfies:
Answer: B — at least one inequality (the union)
Medium Solve the compound inequality \(-3 < x + 2 \le 5\).
Answer: A — \(-5 < x \le 3\); interval \((-5,\,3\,]\)
Medium Solve \(1 \le 2x - 3 < 9\).
Answer: C — \(2 \le x < 6\); interval \([2,\,6)\)
Medium Solve \(x + 4 > 1\) AND \(x - 2 < 3\).
Answer: A — \(-3 < x < 5\); interval \((-3,\,5)\)
Medium Solve \(2x < -6\) OR \(x - 1 > 2\).
Answer: C — \(x < -3\) or \(x > 3\); \((-\infty,\,-3) \cup (3,\,\infty)\)
Hard Solve \(-1 \le -2x + 3 < 7\).
Answer: C — \(-2 < x \le 2\); interval \((-2,\,2\,]\)
Medium Solve \(x > 5\) AND \(x < 2\).
Answer: B — no solution (the pieces don't overlap)
Medium Solve \(x < 4\) OR \(x > 1\).
Answer: B — all real numbers; interval \((-\infty,\,\infty)\)
Medium Write \(-3 \le x < 4\) in interval notation.
Answer: C — \([-3,\,4)\)
Medium Write \(x \le -2\) OR \(x > 6\) in interval notation.
Answer: A — \((-\infty,\,-2\,] \cup (6,\,\infty)\)
Medium Which connective typically produces a union of two separate intervals (a \(\cup\))?
Answer: A — OR (disjunction)
Medium Which of these values satisfy \(-2 \le x < 3\)? Select all that apply.
Answer: C, D
Hard Solve \(-8 < 3x + 1 \le 7\) and give interval notation.
Answer: A — \(-3 < x \le 2\); interval \((-3,\,2\,]\)