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➕ Algebra 1 · Inequalities

Compound Inequalities: notes and common mistakes

Study notes

Key ideas

  • A compound inequality joins two inequalities with “and” or “or”. “And” means both are true. “Or” means at least one is true (both is allowed).
  • a < x ≤ b is shorthand for x > a and x ≤ b. The symbols tell which ends are included.
  • Solve each piece (or do the same move to all three parts), graph, then combine: the overlap for “and”, everything shaded for “or”.
  • An “and” can have no solution, and an “or” can cover every real number. Edge cases: x ≥ c and x ≤ c gives only x = c, and x < c or x > c gives every real number except c.
  • An “or” with two rays that point away from each other cannot be written as a three-part inequality.

Steps

  1. Decide whether the joining word is “and” or “or”. In a word problem, ask whether both limits must hold, or one is enough.
  2. Solve each piece. For a three-part inequality, do the same move to all three parts. If you multiply or divide by a negative number, reverse both signs.
  3. Graph each piece on its own number line.
  4. Combine: the overlap for “and”, everything shaded for “or”.
  5. Write the answer (with the smaller number first in a three-part form), then check a number inside, a number outside, and each boundary.

Formulas

  • Three-part “and” form: a < x ≤ b means x > a and x ≤ b
  • Interval notation: −2 < x ≤ 5 is (−2, 5]. “Or” uses ∪: x < −6 or x ≥ 7 is (−∞, −6) ∪ [7, ∞).
  • Edge cases: x ≥ c and x ≤ c: only x = c. x < c or x > c: every real number except c.

Watch out

  • “Or” is not “between”: x < −2 or x > 5 cannot be written 5 < x < −2.
  • In a three-part inequality, do each move to all three parts, not just the middle.
  • Dividing a three-part inequality by a negative number reverses both signs. Then rewrite it with the smaller number on the left.
  • The symbol on each side decides open or closed: −2 < x ≤ 5 has an open circle at −2 and a closed circle at 5.
  • No overlap in an “and” means no solution. That is a valid answer, not an error.
  • In a word problem, decide between “and” and “or” from the story, and decide whether each end counts (“at least” includes the number, “more than” does not).

Why it works

  • Why is the overlap the answer for “and”? A solution must satisfy both inequalities, and the numbers that satisfy both are exactly the numbers shaded on both number lines.
  • Why do you do the same move to all three parts? A three-part inequality is two inequalities that share the middle expression. Each move must keep both of them true, so it has to be applied to all three parts.
  • Why can an “or” cover every number? When one ray points left and the other points right, and they overlap (as in x < 4 or x > 1), every number is in at least one of them, so nothing is left out. Two rays that point the same way, like x > 2 or x > 7, never cover the whole line.

Common mistakes

✗ Mistakex < −2 or x > 5 can be written 5 < x < −2.
✓ InsteadThat form means x > 5 and x < −2, which no number satisfies. An “or” with two rays pointing away from each other cannot be written as one three-part inequality.
✗ Mistake“Or” means the solution must satisfy both pieces.
✓ Instead“And” means both. “Or” means at least one is enough, so it keeps everything either piece shades.
✗ MistakeTo divide −6 < −3x < 9 by −3, flip the signs and leave the numbers where they are: 2 < x < −3.
✓ InsteadDividing by −3 gives 2 > x > −3. Reading it from right to left puts the smaller number first: −3 < x < 2.
✗ Mistake“Between 2 and 9” always includes 2 and 9.
✓ InsteadIt depends. In math the symbols decide: 2 < x < 9 leaves out both ends, and 2 ≤ x ≤ 9 keeps both. A problem should say “inclusive” when the ends count.
✗ MistakeIf the shaded rays of an “and” do not overlap, I must have made a mistake.
✓ InsteadNo overlap means there is no solution. That is a valid answer, as in x < 1 and x > 4.
✗ MistakeIn −5 < 2x + 1 ≤ 9, I only need to subtract 1 from the middle part.
✓ InsteadDo each move to all three parts: subtract 1 from all of them to get −6 < 2x ≤ 8, then divide all three by 2.

Teach it back

Explain to a friend how to solve 5 ≤ 4x − 3 < 21 and graph it. Then explain how the answer changes if the problem is “4x − 3 < 5 or 4x − 3 ≥ 21”.

Explaining it out loud shows what you really know. Did you…

  • I added 3 to all three parts to get 8 ≤ 4x < 24, then divided all three parts by 4 to get 2 ≤ x < 6.
  • I drew a closed circle at 2, an open circle at 6, and shaded between them. I checked x = 4 (inside) and x = 6 (a boundary that fails).
  • I solved the “or” version as two pieces, x < 2 or x ≥ 6, and said it graphs as two separate rays with a gap, not a stretch in the middle.
  • I explained that “and” keeps the overlap, while “or” keeps everything either piece shades.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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