➕ Algebra 1 · Inequalities
Compound Inequalities: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Finding a time to meet
Mia is free from 3 to 7 in the afternoon, and Sam is free from 5 to 9. Let t be the time in hours after noon. Mia’s free times are 3 ≤ t ≤ 7 and Sam’s are 5 ≤ t ≤ 9. To meet, you need a time when both are free: t must satisfy both inequalities, so 5 ≤ t ≤ 7. If you only need at least one of them to be free, the times are 3 ≤ t ≤ 9.
How it connects: “Both free” is an “and”: the overlap of the two stretches. “At least one free” is an “or”: everything either stretch covers. Here the stretches overlap, so even the “or” answer is one piece. If Mia were free only from 3 to 4, the “and” answer would be empty (no solution) and the “or” answer would be two separate pieces. The limit: the model treats every moment as a possible meeting time, but real meetings also need travel time.
Youth league ages
In a made-up youth league, players must be at least 13 years old and younger than 15. Let a be a player’s age in years. The rule is 13 ≤ a < 15. A player who is exactly 13 can join. A player who is exactly 15 cannot.
How it connects: 13 ≤ a < 15 means a ≥ 13 and a < 15: a closed circle at 13, an open circle at 15, and shading between. “At least” keeps 13 and “younger than” leaves out 15, and the symbols record exactly that. The limit: real leagues usually choose a cutoff date for age, so their exact rules differ from this one.
The two-sided bonus
In a made-up arcade game, you earn a bonus when your final score s is below 100 or at least 900. A score of 50 earns the bonus. A score of 500 earns nothing. A score of exactly 100 earns nothing, and a score of exactly 900 earns the bonus.
How it connects: The rule is s < 100 or s ≥ 900: two separate rays, with an open circle at 100 and a closed circle at 900, and a gap in the middle. It is an “or” because one condition is enough. The limit: scores are whole numbers, so the real picture is separate dots instead of a continuous shaded line. And you cannot squeeze an “or” like this into one three-part inequality.
Goldilocks and the porridge
In the fairy tale “Goldilocks and the Three Bears,” Goldilocks turns down porridge that is too hot and porridge that is too cold, and she eats the bowl that is just right. Suppose, with made-up numbers, that “just right” means a temperature T from 45 to 60 degrees Celsius, including both. Then just right is 45 ≤ T ≤ 60. Too cold is T < 45, and too hot is T > 60. A bowl that is not just right is too cold or too hot.
How it connects: Just right is an “and”: both T ≥ 45 and T ≤ 60 must hold. Not just right is an “or”: T < 45 or T > 60. Notice that the circles swap: 45 and 60 are included in “just right” (closed circles) but not in “too cold” or “too hot” (open circles). The limit: the story gives no numbers, so 45 and 60 are invented for the example.
A mountain trail
A hiking trail climbs steadily from a trailhead at 300 meters above sea level to a summit at 900 meters, with no dips. Let h be your altitude in meters at any point on the climb. You are at the trailhead when h = 300 and at the summit when h = 900, so during the climb, 300 ≤ h ≤ 900.
How it connects: 300 ≤ h ≤ 900 is an “and”: h ≥ 300 and h ≤ 900. Both circles are closed, because you really do stand at the trailhead and at the summit. The limit: the inequality says which altitudes the hike covers, not how long it takes, and it fits only a climb that never goes below the start or above the summit.
You can’t be in two places at once
The saying “you can’t be in two places at once” means that one person cannot be in two separate locations at the same moment.
How it connects: A number is like a person on the number line: it sits in one place. If an “and” asks for a place less than −1 and also a place greater than 2, no number can do both, so there is no solution. The limit: if the two regions overlap, as in x > 1 and x < 4, one number can be in both. And with “or”, being in either place is enough.
Pizza or tacos?
A friend asks a mathematician, “Do you want pizza or tacos?” The mathematician says, “Yes.”
How it connects: In math, “A or B” is true when at least one part is true, including when both are. The mathematician wants at least one of them, so “yes” is a true answer. That is why an “or” inequality keeps everything either piece shades. The limit: in everyday talk, this question asks you to pick one, so “yes” is not a helpful answer.
Memory help
SPLIT
Split a three-part inequality into two pieces (an “or” already comes in two), and solve each piece. Plot each piece on its own number line. Look at the joining word: “and” or “or”. Intersect for “and”: keep only the overlap. Take everything shaded for “or”: keep whatever either piece shades.
And narrows, or widens
Every solution of an “and” has to satisfy both pieces, so “and” can only narrow things down (or keep them the same). An “or” needs just one piece, so it can only widen them (or keep them the same). If your “and” answer is bigger than one of its pieces, or your “or” answer is smaller than one of its pieces, recheck.
Two highlighters
Highlight the first piece in yellow and the second in blue. Where they overlap, the page looks green. For “and”, keep only the green part. For “or”, keep every part that is yellow, blue or green.
“Or” is not “between”
x < −2 or x > 5 cannot be written 5 < x < −2. That one-line form means x > 5 and x < −2, which no number satisfies. Only an “and” can be squeezed into a three-part inequality.
All three parts, and flip them all
In a three-part inequality, every move goes to all three parts, not just the middle. If you multiply or divide by a negative number, reverse both signs. If the larger number is then on the left, write the chain in the opposite order (each symbol turns to face the other way) so the smaller number comes first: −1 > x ≥ −4 is the same as −4 ≤ x < −1.
Read it out loud
Say each inequality as a sentence before you solve. “−2 < x ≤ 5” is “x is greater than −2 and at most 5”. “x < −6 or x ≥ 7” is “x is less than −6, or x is at least 7”. If the sentence has “and”, think overlap. If it has “or”, think either.